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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_000006", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_000009", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_000281", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_000382", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): The standard deviation is the square root of the variance."}]}, "response": "TRUE — The standard deviation is the square root of the variance.", "meta": {"id": "WUA_STAT_25K_000383", "domain": "master_scholars_academics", "subdomain": "stats::distribution", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): The standard deviation is the square root of the variance."}]}, "response": "TRUE — The standard deviation is the square root of the variance.", "meta": {"id": "WUA_STAT_25K_000384", "domain": "master_scholars_academics", "subdomain": "stats::distribution", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_000386", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_000387", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_000388", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_000389", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_000390", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: The standard deviation is the square root of the variance."}]}, "response": "TRUE — The standard deviation is the square root of the variance.", "meta": {"id": "WUA_STAT_25K_000420", "domain": "master_scholars_academics", "subdomain": "stats::distribution", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_000424", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_000425", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_000876", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_001381", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_001614", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A probability must lie between 0 and 1 inclusive."}]}, "response": "TRUE — A probability must lie between 0 and 1 inclusive.", "meta": {"id": "WUA_STAT_25K_001900", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_001904", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_002005", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_002006", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_002008", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_002009", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_002010", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_002012", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_002013", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_002014", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_002098", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_002226", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_002815", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_002816", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: The standard deviation is the square root of the variance."}]}, "response": "TRUE — The standard deviation is the square root of the variance.", "meta": {"id": "WUA_STAT_25K_002885", "domain": "master_scholars_academics", "subdomain": "stats::distribution", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_002886", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_002892", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_003447", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_003914", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_003916", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): The standard deviation is the square root of the variance."}]}, "response": "TRUE — The standard deviation is the square root of the variance.", "meta": {"id": "WUA_STAT_25K_003918", "domain": "master_scholars_academics", "subdomain": "stats::distribution", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: The standard deviation is the square root of the variance."}]}, "response": "TRUE — The standard deviation is the square root of the variance.", "meta": {"id": "WUA_STAT_25K_004145", "domain": "master_scholars_academics", "subdomain": "stats::distribution", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: The standard deviation is the square root of the variance."}]}, "response": "TRUE — The standard deviation is the square root of the variance.", "meta": {"id": "WUA_STAT_25K_004151", "domain": "master_scholars_academics", "subdomain": "stats::distribution", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_004476", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_004505", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): The standard deviation is the square root of the variance."}]}, "response": "TRUE — The standard deviation is the square root of the variance.", "meta": {"id": "WUA_STAT_25K_004525", "domain": "master_scholars_academics", "subdomain": "stats::distribution", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_005333", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_006149", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_006775", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_006778", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_006779", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_006793", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_006794", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_007351", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: The standard deviation is the square root of the variance."}]}, "response": "TRUE — The standard deviation is the square root of the variance.", "meta": {"id": "WUA_STAT_25K_007445", "domain": "master_scholars_academics", "subdomain": "stats::distribution", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_007992", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_007993", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_007994", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: Independence means P(A∩B) = P(A)P(B)."}]}, "response": "TRUE — Independence means P(A∩B) = P(A)P(B).", "meta": {"id": "WUA_STAT_25K_007995", "domain": "master_scholars_academics", "subdomain": "stats::probability", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_007996", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_007997", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "core", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Academic fact-check: TRUE/FALSE then corrected statement if needed: A p-value is the probability that the null hypothesis is true."}]}, "response": "FALSE — A p-value is P(data as or more extreme | null), not P(null | data).", "meta": {"id": "WUA_STAT_25K_007998", "domain": "master_scholars_academics", "subdomain": "stats::hypothesis_testing", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "Verify TRUE/FALSE and correct if false: Correlation implies causation."}]}, "response": "FALSE — Correlation alone does not establish causation.", "meta": {"id": "WUA_STAT_25K_007999", "domain": "master_scholars_academics", "subdomain": "stats::modeling", "task": "fact_check", "difficulty": "seed", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide concise, factual answers."}, {"role": "user", "content": "True or False (academics): The standard deviation is the square root of the variance."}]}, "response": "TRUE — The standard deviation is the square root of the variance.", "meta": {"id": "WUA_STAT_25K_008000", "domain": "master_scholars_academics", "subdomain": "stats::distribution", "task": "fact_check", "difficulty": "advanced", "truth_mode": "verifiable_fact", "wrapper": "verify_true_false", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "verification"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.48, s=35.25, n=194, compute CI."}]}, "response": "95% CI ≈ [87.52, 97.44].", "meta": {"id": "WUA_STAT_25K_008001", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.59, compute E and Var."}]}, "response": "E[X]=20.65; Var[X]=8.466.", "meta": {"id": "WUA_STAT_25K_008002", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.7, compute E and Var."}]}, "response": "E[X]=12.60; Var[X]=3.780.", "meta": {"id": "WUA_STAT_25K_008003", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.84, μ=58.28, σ=3.93, compute z."}]}, "response": "z = -12.835.", "meta": {"id": "WUA_STAT_25K_008004", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=192.05, s=28.61, n=134, compute CI."}]}, "response": "95% CI ≈ [187.21, 196.89].", "meta": {"id": "WUA_STAT_25K_008005", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.85, s=8.32, n=191, compute CI."}]}, "response": "95% CI ≈ [110.67, 113.03].", "meta": {"id": "WUA_STAT_25K_008006", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.27, μ=46.77, σ=2.23, compute z."}]}, "response": "z = 8.744.", "meta": {"id": "WUA_STAT_25K_008007", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.064, sens=0.94, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.682.", "meta": {"id": "WUA_STAT_25K_008008", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.73, s=29.34, n=93, compute CI."}]}, "response": "95% CI ≈ [63.77, 75.69].", "meta": {"id": "WUA_STAT_25K_008009", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.29, s=28.89, n=48, compute CI."}]}, "response": "95% CI ≈ [101.12, 117.46].", "meta": {"id": "WUA_STAT_25K_008010", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.38, μ=36.33, σ=1.95, compute z."}]}, "response": "z = 25.667.", "meta": {"id": "WUA_STAT_25K_008011", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.86, s=30.09, n=94, compute CI."}]}, "response": "95% CI ≈ [153.78, 165.94].", "meta": {"id": "WUA_STAT_25K_008012", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=200.0, s=21.52, n=104, compute CI."}]}, "response": "95% CI ≈ [195.86, 204.14].", "meta": {"id": "WUA_STAT_25K_008013", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.74, μ=25.16, σ=21.3, compute z."}]}, "response": "z = 0.262.", "meta": {"id": "WUA_STAT_25K_008014", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.46, μ=23.86, σ=13.35, compute z."}]}, "response": "z = -1.079.", "meta": {"id": "WUA_STAT_25K_008015", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.17, s=10.68, n=35, compute CI."}]}, "response": "95% CI ≈ [136.63, 143.71].", "meta": {"id": "WUA_STAT_25K_008016", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.04, μ=37.57, σ=3.37, compute z."}]}, "response": "z = -9.356.", "meta": {"id": "WUA_STAT_25K_008017", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.93, μ=32.96, σ=16.17, compute z."}]}, "response": "z = 0.431.", "meta": {"id": "WUA_STAT_25K_008018", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.42, compute E and Var."}]}, "response": "E[X]=4.62; Var[X]=2.680.", "meta": {"id": "WUA_STAT_25K_008019", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.68, μ=64.83, σ=13.32, compute z."}]}, "response": "z = 2.466.", "meta": {"id": "WUA_STAT_25K_008020", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.79, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.493.", "meta": {"id": "WUA_STAT_25K_008021", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.29, μ=80.16, σ=4.62, compute z."}]}, "response": "z = -3.219.", "meta": {"id": "WUA_STAT_25K_008022", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.88, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.690.", "meta": {"id": "WUA_STAT_25K_008023", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.9, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.448.", "meta": {"id": "WUA_STAT_25K_008024", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=72.36, s=26.14, n=82, compute CI."}]}, "response": "95% CI ≈ [66.70, 78.02].", "meta": {"id": "WUA_STAT_25K_008025", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.43, μ=89.36, σ=17.06, compute z."}]}, "response": "z = -0.758.", "meta": {"id": "WUA_STAT_25K_008026", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=103.36, s=10.81, n=101, compute CI."}]}, "response": "95% CI ≈ [101.25, 105.47].", "meta": {"id": "WUA_STAT_25K_008027", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.12, s=32.91, n=156, compute CI."}]}, "response": "95% CI ≈ [37.96, 48.28].", "meta": {"id": "WUA_STAT_25K_008028", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.84, s=36.49, n=78, compute CI."}]}, "response": "95% CI ≈ [80.74, 96.94].", "meta": {"id": "WUA_STAT_25K_008029", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.052, sens=0.83, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.477.", "meta": {"id": "WUA_STAT_25K_008030", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.59, s=19.6, n=187, compute CI."}]}, "response": "95% CI ≈ [167.78, 173.40].", "meta": {"id": "WUA_STAT_25K_008031", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.117, sens=0.98, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.309.", "meta": {"id": "WUA_STAT_25K_008032", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.87, μ=99.93, σ=4.7, compute z."}]}, "response": "z = -6.821.", "meta": {"id": "WUA_STAT_25K_008033", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.51, compute E and Var."}]}, "response": "E[X]=11.73; Var[X]=5.748.", "meta": {"id": "WUA_STAT_25K_008034", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.65, s=9.66, n=67, compute CI."}]}, "response": "95% CI ≈ [133.34, 137.96].", "meta": {"id": "WUA_STAT_25K_008035", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=47.41, s=7.09, n=82, compute CI."}]}, "response": "95% CI ≈ [45.88, 48.94].", "meta": {"id": "WUA_STAT_25K_008036", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.85, μ=55.47, σ=6.75, compute z."}]}, "response": "z = 5.686.", "meta": {"id": "WUA_STAT_25K_008037", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.02, μ=26.68, σ=15.6, compute z."}]}, "response": "z = 0.406.", "meta": {"id": "WUA_STAT_25K_008038", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.46, s=19.54, n=95, compute CI."}]}, "response": "95% CI ≈ [130.53, 138.39].", "meta": {"id": "WUA_STAT_25K_008039", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=96.07, s=8.76, n=162, compute CI."}]}, "response": "95% CI ≈ [94.72, 97.42].", "meta": {"id": "WUA_STAT_25K_008040", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.83, s=11.89, n=141, compute CI."}]}, "response": "95% CI ≈ [48.87, 52.79].", "meta": {"id": "WUA_STAT_25K_008041", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.82, μ=17.31, σ=3.56, compute z."}]}, "response": "z = 12.222.", "meta": {"id": "WUA_STAT_25K_008042", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.71, compute E and Var."}]}, "response": "E[X]=11.36; Var[X]=3.294.", "meta": {"id": "WUA_STAT_25K_008043", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.98, s=31.65, n=133, compute CI."}]}, "response": "95% CI ≈ [181.60, 192.36].", "meta": {"id": "WUA_STAT_25K_008044", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.74, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.770.", "meta": {"id": "WUA_STAT_25K_008045", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.4, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=5.040.", "meta": {"id": "WUA_STAT_25K_008046", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.95, s=9.73, n=194, compute CI."}]}, "response": "95% CI ≈ [84.58, 87.32].", "meta": {"id": "WUA_STAT_25K_008047", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.93, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.415.", "meta": {"id": "WUA_STAT_25K_008048", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.87, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.523.", "meta": {"id": "WUA_STAT_25K_008049", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.059, sens=0.99, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.280.", "meta": {"id": "WUA_STAT_25K_008050", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.8, μ=96.26, σ=14.67, compute z."}]}, "response": "z = 0.173.", "meta": {"id": "WUA_STAT_25K_008051", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.94, μ=59.64, σ=1.67, compute z."}]}, "response": "z = 7.964.", "meta": {"id": "WUA_STAT_25K_008052", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=10.4, s=29.85, n=183, compute CI."}]}, "response": "95% CI ≈ [6.08, 14.72].", "meta": {"id": "WUA_STAT_25K_008053", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.52, s=26.35, n=68, compute CI."}]}, "response": "95% CI ≈ [106.26, 118.78].", "meta": {"id": "WUA_STAT_25K_008054", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.6, compute E and Var."}]}, "response": "E[X]=6.00; Var[X]=2.400.", "meta": {"id": "WUA_STAT_25K_008055", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.67, μ=74.36, σ=15.71, compute z."}]}, "response": "z = -0.489.", "meta": {"id": "WUA_STAT_25K_008056", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.75, compute E and Var."}]}, "response": "E[X]=9.00; Var[X]=2.250.", "meta": {"id": "WUA_STAT_25K_008057", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.57, compute E and Var."}]}, "response": "E[X]=5.70; Var[X]=2.451.", "meta": {"id": "WUA_STAT_25K_008058", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.74, compute E and Var."}]}, "response": "E[X]=22.94; Var[X]=5.964.", "meta": {"id": "WUA_STAT_25K_008059", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.61, s=24.48, n=151, compute CI."}]}, "response": "95% CI ≈ [156.71, 164.51].", "meta": {"id": "WUA_STAT_25K_008060", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.65, compute E and Var."}]}, "response": "E[X]=3.25; Var[X]=1.137.", "meta": {"id": "WUA_STAT_25K_008061", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=95.49, s=35.8, n=194, compute CI."}]}, "response": "95% CI ≈ [90.45, 100.53].", "meta": {"id": "WUA_STAT_25K_008062", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.8, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.261.", "meta": {"id": "WUA_STAT_25K_008063", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.57, compute E and Var."}]}, "response": "E[X]=12.54; Var[X]=5.392.", "meta": {"id": "WUA_STAT_25K_008064", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.79, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.263.", "meta": {"id": "WUA_STAT_25K_008065", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.44, compute E and Var."}]}, "response": "E[X]=4.84; Var[X]=2.710.", "meta": {"id": "WUA_STAT_25K_008066", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.77, s=34.35, n=101, compute CI."}]}, "response": "95% CI ≈ [171.07, 184.47].", "meta": {"id": "WUA_STAT_25K_008067", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=115.04, s=19.04, n=88, compute CI."}]}, "response": "95% CI ≈ [111.06, 119.02].", "meta": {"id": "WUA_STAT_25K_008068", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.95, μ=97.16, σ=7.44, compute z."}]}, "response": "z = -11.722.", "meta": {"id": "WUA_STAT_25K_008069", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.33, s=26.13, n=81, compute CI."}]}, "response": "95% CI ≈ [5.64, 17.02].", "meta": {"id": "WUA_STAT_25K_008070", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=20.91, s=38.78, n=30, compute CI."}]}, "response": "95% CI ≈ [7.03, 34.79].", "meta": {"id": "WUA_STAT_25K_008071", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.27, s=21.64, n=45, compute CI."}]}, "response": "95% CI ≈ [11.95, 24.59].", "meta": {"id": "WUA_STAT_25K_008072", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=103.48, s=29.8, n=65, compute CI."}]}, "response": "95% CI ≈ [96.24, 110.72].", "meta": {"id": "WUA_STAT_25K_008073", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.034, sens=0.98, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.633.", "meta": {"id": "WUA_STAT_25K_008074", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.32, μ=67.88, σ=15.05, compute z."}]}, "response": "z = 0.827.", "meta": {"id": "WUA_STAT_25K_008075", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.06, compute E and Var."}]}, "response": "E[X]=1.62; Var[X]=1.523.", "meta": {"id": "WUA_STAT_25K_008076", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.17, μ=64.71, σ=11.42, compute z."}]}, "response": "z = -4.863.", "meta": {"id": "WUA_STAT_25K_008077", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.39, μ=33.15, σ=24.67, compute z."}]}, "response": "z = 2.604.", "meta": {"id": "WUA_STAT_25K_008078", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.86, μ=78.03, σ=3.38, compute z."}]}, "response": "z = -1.530.", "meta": {"id": "WUA_STAT_25K_008079", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.85, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.358.", "meta": {"id": "WUA_STAT_25K_008080", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=128.42, s=13.67, n=29, compute CI."}]}, "response": "95% CI ≈ [123.44, 133.40].", "meta": {"id": "WUA_STAT_25K_008081", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.72, μ=74.12, σ=18.82, compute z."}]}, "response": "z = -3.900.", "meta": {"id": "WUA_STAT_25K_008082", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.13, compute E and Var."}]}, "response": "E[X]=0.91; Var[X]=0.792.", "meta": {"id": "WUA_STAT_25K_008083", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.66, μ=97.88, σ=24.2, compute z."}]}, "response": "z = -2.075.", "meta": {"id": "WUA_STAT_25K_008084", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.97, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.445.", "meta": {"id": "WUA_STAT_25K_008085", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=93.53, s=20.92, n=32, compute CI."}]}, "response": "95% CI ≈ [86.28, 100.78].", "meta": {"id": "WUA_STAT_25K_008086", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.23, compute E and Var."}]}, "response": "E[X]=8.51; Var[X]=6.553.", "meta": {"id": "WUA_STAT_25K_008087", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.5, s=36.53, n=132, compute CI."}]}, "response": "95% CI ≈ [181.27, 193.73].", "meta": {"id": "WUA_STAT_25K_008088", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.01, s=16.37, n=58, compute CI."}]}, "response": "95% CI ≈ [94.80, 103.22].", "meta": {"id": "WUA_STAT_25K_008089", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=32.67, s=22.33, n=163, compute CI."}]}, "response": "95% CI ≈ [29.24, 36.10].", "meta": {"id": "WUA_STAT_25K_008090", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.44, s=32.97, n=88, compute CI."}]}, "response": "95% CI ≈ [187.55, 201.33].", "meta": {"id": "WUA_STAT_25K_008091", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.187, sens=0.96, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.501.", "meta": {"id": "WUA_STAT_25K_008092", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=100.94, s=16.15, n=150, compute CI."}]}, "response": "95% CI ≈ [98.36, 103.52].", "meta": {"id": "WUA_STAT_25K_008093", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.38, μ=59.22, σ=10.09, compute z."}]}, "response": "z = -0.480.", "meta": {"id": "WUA_STAT_25K_008094", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.11, μ=15.08, σ=19.97, compute z."}]}, "response": "z = 2.605.", "meta": {"id": "WUA_STAT_25K_008095", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=95.89, s=33.79, n=197, compute CI."}]}, "response": "95% CI ≈ [91.17, 100.61].", "meta": {"id": "WUA_STAT_25K_008096", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.021, sens=0.86, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.480.", "meta": {"id": "WUA_STAT_25K_008097", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.28, μ=61.5, σ=8.87, compute z."}]}, "response": "z = 2.794.", "meta": {"id": "WUA_STAT_25K_008098", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.87, μ=88.4, σ=19.18, compute z."}]}, "response": "z = -0.445.", "meta": {"id": "WUA_STAT_25K_008099", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.02, μ=67.47, σ=2.47, compute z."}]}, "response": "z = 0.223.", "meta": {"id": "WUA_STAT_25K_008100", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.095, sens=0.72, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.716.", "meta": {"id": "WUA_STAT_25K_008101", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.21, s=15.13, n=119, compute CI."}]}, "response": "95% CI ≈ [128.49, 133.93].", "meta": {"id": "WUA_STAT_25K_008102", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.084, sens=0.84, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.391.", "meta": {"id": "WUA_STAT_25K_008103", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.01, s=14.97, n=193, compute CI."}]}, "response": "95% CI ≈ [120.90, 125.12].", "meta": {"id": "WUA_STAT_25K_008104", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.229, sens=0.88, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.592.", "meta": {"id": "WUA_STAT_25K_008105", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.148, sens=0.75, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.685.", "meta": {"id": "WUA_STAT_25K_008106", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.51, compute E and Var."}]}, "response": "E[X]=16.32; Var[X]=7.997.", "meta": {"id": "WUA_STAT_25K_008107", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.41, compute E and Var."}]}, "response": "E[X]=2.46; Var[X]=1.451.", "meta": {"id": "WUA_STAT_25K_008108", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.41, compute E and Var."}]}, "response": "E[X]=2.87; Var[X]=1.693.", "meta": {"id": "WUA_STAT_25K_008109", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.46, compute E and Var."}]}, "response": "E[X]=11.96; Var[X]=6.458.", "meta": {"id": "WUA_STAT_25K_008110", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.04, s=16.85, n=84, compute CI."}]}, "response": "95% CI ≈ [153.44, 160.64].", "meta": {"id": "WUA_STAT_25K_008111", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.57, compute E and Var."}]}, "response": "E[X]=7.98; Var[X]=3.431.", "meta": {"id": "WUA_STAT_25K_008112", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.2, μ=1.96, σ=24.07, compute z."}]}, "response": "z = 0.425.", "meta": {"id": "WUA_STAT_25K_008113", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.15, s=5.9, n=131, compute CI."}]}, "response": "95% CI ≈ [85.14, 87.16].", "meta": {"id": "WUA_STAT_25K_008114", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.42, μ=47.95, σ=3.0, compute z."}]}, "response": "z = -15.177.", "meta": {"id": "WUA_STAT_25K_008115", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.13, compute E and Var."}]}, "response": "E[X]=4.68; Var[X]=4.072.", "meta": {"id": "WUA_STAT_25K_008116", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.25, compute E and Var."}]}, "response": "E[X]=7.25; Var[X]=5.438.", "meta": {"id": "WUA_STAT_25K_008117", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.51, μ=49.9, σ=10.19, compute z."}]}, "response": "z = -3.375.", "meta": {"id": "WUA_STAT_25K_008118", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.97, μ=16.67, σ=9.26, compute z."}]}, "response": "z = 3.056.", "meta": {"id": "WUA_STAT_25K_008119", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.02, μ=20.72, σ=13.77, compute z."}]}, "response": "z = -1.213.", "meta": {"id": "WUA_STAT_25K_008120", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.186, sens=0.77, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.687.", "meta": {"id": "WUA_STAT_25K_008121", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.3, compute E and Var."}]}, "response": "E[X]=3.90; Var[X]=2.730.", "meta": {"id": "WUA_STAT_25K_008122", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.54, compute E and Var."}]}, "response": "E[X]=20.52; Var[X]=9.439.", "meta": {"id": "WUA_STAT_25K_008123", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.94, s=28.49, n=165, compute CI."}]}, "response": "95% CI ≈ [117.59, 126.29].", "meta": {"id": "WUA_STAT_25K_008124", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.95, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.935.", "meta": {"id": "WUA_STAT_25K_008125", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.74, compute E and Var."}]}, "response": "E[X]=24.42; Var[X]=6.349.", "meta": {"id": "WUA_STAT_25K_008126", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.62, compute E and Var."}]}, "response": "E[X]=7.44; Var[X]=2.827.", "meta": {"id": "WUA_STAT_25K_008127", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.9, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.867.", "meta": {"id": "WUA_STAT_25K_008128", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.38, μ=16.57, σ=22.11, compute z."}]}, "response": "z = -0.144.", "meta": {"id": "WUA_STAT_25K_008129", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.173, sens=0.87, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.752.", "meta": {"id": "WUA_STAT_25K_008130", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.39, compute E and Var."}]}, "response": "E[X]=12.09; Var[X]=7.375.", "meta": {"id": "WUA_STAT_25K_008131", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.72, μ=99.1, σ=18.64, compute z."}]}, "response": "z = -2.595.", "meta": {"id": "WUA_STAT_25K_008132", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.66, s=18.18, n=114, compute CI."}]}, "response": "95% CI ≈ [149.32, 156.00].", "meta": {"id": "WUA_STAT_25K_008133", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.93, s=30.63, n=78, compute CI."}]}, "response": "95% CI ≈ [72.13, 85.73].", "meta": {"id": "WUA_STAT_25K_008134", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.29, compute E and Var."}]}, "response": "E[X]=10.73; Var[X]=7.618.", "meta": {"id": "WUA_STAT_25K_008135", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.8, s=39.93, n=103, compute CI."}]}, "response": "95% CI ≈ [5.09, 20.51].", "meta": {"id": "WUA_STAT_25K_008136", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.16, compute E and Var."}]}, "response": "E[X]=4.16; Var[X]=3.494.", "meta": {"id": "WUA_STAT_25K_008137", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.71, compute E and Var."}]}, "response": "E[X]=18.46; Var[X]=5.353.", "meta": {"id": "WUA_STAT_25K_008138", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.49, μ=81.89, σ=18.33, compute z."}]}, "response": "z = -4.441.", "meta": {"id": "WUA_STAT_25K_008139", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.048, sens=0.86, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.302.", "meta": {"id": "WUA_STAT_25K_008140", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.07, sens=0.84, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.760.", "meta": {"id": "WUA_STAT_25K_008141", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.75, s=39.52, n=67, compute CI."}]}, "response": "95% CI ≈ [75.29, 94.21].", "meta": {"id": "WUA_STAT_25K_008142", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.23, s=34.98, n=129, compute CI."}]}, "response": "95% CI ≈ [140.19, 152.27].", "meta": {"id": "WUA_STAT_25K_008143", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.69, μ=41.53, σ=2.46, compute z."}]}, "response": "z = -12.537.", "meta": {"id": "WUA_STAT_25K_008144", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.0, s=7.3, n=71, compute CI."}]}, "response": "95% CI ≈ [134.30, 137.70].", "meta": {"id": "WUA_STAT_25K_008145", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.34, μ=39.35, σ=24.87, compute z."}]}, "response": "z = -0.925.", "meta": {"id": "WUA_STAT_25K_008146", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.2, s=10.99, n=110, compute CI."}]}, "response": "95% CI ≈ [84.15, 88.25].", "meta": {"id": "WUA_STAT_25K_008147", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.46, s=37.65, n=99, compute CI."}]}, "response": "95% CI ≈ [68.04, 82.88].", "meta": {"id": "WUA_STAT_25K_008148", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.49, μ=29.99, σ=4.67, compute z."}]}, "response": "z = -5.032.", "meta": {"id": "WUA_STAT_25K_008149", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.23, μ=56.28, σ=6.97, compute z."}]}, "response": "z = 0.997.", "meta": {"id": "WUA_STAT_25K_008150", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.17, μ=4.36, σ=15.66, compute z."}]}, "response": "z = 2.798.", "meta": {"id": "WUA_STAT_25K_008151", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.92, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.888.", "meta": {"id": "WUA_STAT_25K_008152", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.91, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.683.", "meta": {"id": "WUA_STAT_25K_008153", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.76, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.331.", "meta": {"id": "WUA_STAT_25K_008154", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.35, s=10.13, n=70, compute CI."}]}, "response": "95% CI ≈ [51.98, 56.72].", "meta": {"id": "WUA_STAT_25K_008155", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.48, s=7.6, n=105, compute CI."}]}, "response": "95% CI ≈ [193.03, 195.93].", "meta": {"id": "WUA_STAT_25K_008156", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.78, s=10.46, n=37, compute CI."}]}, "response": "95% CI ≈ [167.41, 174.15].", "meta": {"id": "WUA_STAT_25K_008157", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.4, compute E and Var."}]}, "response": "E[X]=12.40; Var[X]=7.440.", "meta": {"id": "WUA_STAT_25K_008158", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.16, s=23.46, n=144, compute CI."}]}, "response": "95% CI ≈ [29.33, 36.99].", "meta": {"id": "WUA_STAT_25K_008159", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.57, compute E and Var."}]}, "response": "E[X]=18.81; Var[X]=8.088.", "meta": {"id": "WUA_STAT_25K_008160", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.6, compute E and Var."}]}, "response": "E[X]=10.20; Var[X]=4.080.", "meta": {"id": "WUA_STAT_25K_008161", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.66, μ=21.94, σ=18.75, compute z."}]}, "response": "z = 4.038.", "meta": {"id": "WUA_STAT_25K_008162", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.03, μ=68.15, σ=10.99, compute z."}]}, "response": "z = 2.628.", "meta": {"id": "WUA_STAT_25K_008163", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.59, μ=75.34, σ=10.37, compute z."}]}, "response": "z = -2.580.", "meta": {"id": "WUA_STAT_25K_008164", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.17, compute E and Var."}]}, "response": "E[X]=4.42; Var[X]=3.669.", "meta": {"id": "WUA_STAT_25K_008165", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.87, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.478.", "meta": {"id": "WUA_STAT_25K_008166", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.55, s=2.72, n=142, compute CI."}]}, "response": "95% CI ≈ [57.10, 58.00].", "meta": {"id": "WUA_STAT_25K_008167", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.26, compute E and Var."}]}, "response": "E[X]=5.72; Var[X]=4.233.", "meta": {"id": "WUA_STAT_25K_008168", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.65, compute E and Var."}]}, "response": "E[X]=21.45; Var[X]=7.507.", "meta": {"id": "WUA_STAT_25K_008169", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.55, compute E and Var."}]}, "response": "E[X]=15.95; Var[X]=7.178.", "meta": {"id": "WUA_STAT_25K_008170", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=39.46, s=32.78, n=165, compute CI."}]}, "response": "95% CI ≈ [34.46, 44.46].", "meta": {"id": "WUA_STAT_25K_008171", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.23, compute E and Var."}]}, "response": "E[X]=2.99; Var[X]=2.302.", "meta": {"id": "WUA_STAT_25K_008172", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.25, μ=93.76, σ=22.46, compute z."}]}, "response": "z = -4.163.", "meta": {"id": "WUA_STAT_25K_008173", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.81, μ=20.5, σ=3.31, compute z."}]}, "response": "z = 1.000.", "meta": {"id": "WUA_STAT_25K_008174", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.047, sens=0.86, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.459.", "meta": {"id": "WUA_STAT_25K_008175", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.82, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.432.", "meta": {"id": "WUA_STAT_25K_008176", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.18, compute E and Var."}]}, "response": "E[X]=4.68; Var[X]=3.838.", "meta": {"id": "WUA_STAT_25K_008177", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.27, s=22.64, n=167, compute CI."}]}, "response": "95% CI ≈ [41.84, 48.70].", "meta": {"id": "WUA_STAT_25K_008178", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.96, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.314.", "meta": {"id": "WUA_STAT_25K_008179", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.58, compute E and Var."}]}, "response": "E[X]=22.62; Var[X]=9.500.", "meta": {"id": "WUA_STAT_25K_008180", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.43, μ=83.89, σ=1.75, compute z."}]}, "response": "z = -23.120.", "meta": {"id": "WUA_STAT_25K_008181", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.105, sens=0.8, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.245.", "meta": {"id": "WUA_STAT_25K_008182", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.54, s=29.44, n=117, compute CI."}]}, "response": "95% CI ≈ [96.21, 106.87].", "meta": {"id": "WUA_STAT_25K_008183", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.24, compute E and Var."}]}, "response": "E[X]=2.16; Var[X]=1.642.", "meta": {"id": "WUA_STAT_25K_008184", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.43, s=24.61, n=65, compute CI."}]}, "response": "95% CI ≈ [40.45, 52.41].", "meta": {"id": "WUA_STAT_25K_008185", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.68, s=38.53, n=182, compute CI."}]}, "response": "95% CI ≈ [93.08, 104.28].", "meta": {"id": "WUA_STAT_25K_008186", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.44, s=27.43, n=126, compute CI."}]}, "response": "95% CI ≈ [154.65, 164.23].", "meta": {"id": "WUA_STAT_25K_008187", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.99, s=22.85, n=22, compute CI."}]}, "response": "95% CI ≈ [151.44, 170.54].", "meta": {"id": "WUA_STAT_25K_008188", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.09, compute E and Var."}]}, "response": "E[X]=1.98; Var[X]=1.802.", "meta": {"id": "WUA_STAT_25K_008189", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.05, sens=0.99, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.343.", "meta": {"id": "WUA_STAT_25K_008190", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.55, compute E and Var."}]}, "response": "E[X]=4.40; Var[X]=1.980.", "meta": {"id": "WUA_STAT_25K_008191", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.94, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.790.", "meta": {"id": "WUA_STAT_25K_008192", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.185, sens=0.87, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.473.", "meta": {"id": "WUA_STAT_25K_008193", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.67, μ=93.76, σ=15.76, compute z."}]}, "response": "z = -2.417.", "meta": {"id": "WUA_STAT_25K_008194", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.13, compute E and Var."}]}, "response": "E[X]=4.03; Var[X]=3.506.", "meta": {"id": "WUA_STAT_25K_008195", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.41, compute E and Var."}]}, "response": "E[X]=4.51; Var[X]=2.661.", "meta": {"id": "WUA_STAT_25K_008196", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.87, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.379.", "meta": {"id": "WUA_STAT_25K_008197", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.96, μ=88.79, σ=10.39, compute z."}]}, "response": "z = -4.026.", "meta": {"id": "WUA_STAT_25K_008198", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.01, μ=78.45, σ=14.34, compute z."}]}, "response": "z = -2.890.", "meta": {"id": "WUA_STAT_25K_008199", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=68.37, s=39.92, n=188, compute CI."}]}, "response": "95% CI ≈ [62.66, 74.08].", "meta": {"id": "WUA_STAT_25K_008200", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.75, μ=62.78, σ=19.65, compute z."}]}, "response": "z = 0.813.", "meta": {"id": "WUA_STAT_25K_008201", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.85, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.423.", "meta": {"id": "WUA_STAT_25K_008202", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.36, compute E and Var."}]}, "response": "E[X]=3.60; Var[X]=2.304.", "meta": {"id": "WUA_STAT_25K_008203", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.79, compute E and Var."}]}, "response": "E[X]=22.91; Var[X]=4.811.", "meta": {"id": "WUA_STAT_25K_008204", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.93, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.311.", "meta": {"id": "WUA_STAT_25K_008205", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.93, μ=70.71, σ=2.35, compute z."}]}, "response": "z = 0.945.", "meta": {"id": "WUA_STAT_25K_008206", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.96, μ=10.29, σ=21.85, compute z."}]}, "response": "z = 3.463.", "meta": {"id": "WUA_STAT_25K_008207", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.01, μ=66.59, σ=6.34, compute z."}]}, "response": "z = -6.874.", "meta": {"id": "WUA_STAT_25K_008208", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.248, sens=0.91, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.769.", "meta": {"id": "WUA_STAT_25K_008209", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.34, compute E and Var."}]}, "response": "E[X]=3.74; Var[X]=2.468.", "meta": {"id": "WUA_STAT_25K_008210", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.77, μ=46.59, σ=14.67, compute z."}]}, "response": "z = -2.101.", "meta": {"id": "WUA_STAT_25K_008211", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.76, compute E and Var."}]}, "response": "E[X]=21.28; Var[X]=5.107.", "meta": {"id": "WUA_STAT_25K_008212", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.79, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.454.", "meta": {"id": "WUA_STAT_25K_008213", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.4, s=13.22, n=104, compute CI."}]}, "response": "95% CI ≈ [71.86, 76.94].", "meta": {"id": "WUA_STAT_25K_008214", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.56, s=21.62, n=28, compute CI."}]}, "response": "95% CI ≈ [181.55, 197.57].", "meta": {"id": "WUA_STAT_25K_008215", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.67, compute E and Var."}]}, "response": "E[X]=24.12; Var[X]=7.960.", "meta": {"id": "WUA_STAT_25K_008216", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.059, sens=0.7, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.385.", "meta": {"id": "WUA_STAT_25K_008217", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=16.83, s=32.8, n=76, compute CI."}]}, "response": "95% CI ≈ [9.46, 24.20].", "meta": {"id": "WUA_STAT_25K_008218", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.63, s=12.22, n=145, compute CI."}]}, "response": "95% CI ≈ [11.64, 15.62].", "meta": {"id": "WUA_STAT_25K_008219", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.49, μ=9.41, σ=23.05, compute z."}]}, "response": "z = 1.956.", "meta": {"id": "WUA_STAT_25K_008220", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.93, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.115.", "meta": {"id": "WUA_STAT_25K_008221", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.93, μ=21.03, σ=23.74, compute z."}]}, "response": "z = 0.417.", "meta": {"id": "WUA_STAT_25K_008222", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.162, sens=0.83, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.616.", "meta": {"id": "WUA_STAT_25K_008223", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.25, compute E and Var."}]}, "response": "E[X]=1.25; Var[X]=0.938.", "meta": {"id": "WUA_STAT_25K_008224", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.52, compute E and Var."}]}, "response": "E[X]=14.04; Var[X]=6.739.", "meta": {"id": "WUA_STAT_25K_008225", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.73, s=12.11, n=135, compute CI."}]}, "response": "95% CI ≈ [47.69, 51.77].", "meta": {"id": "WUA_STAT_25K_008226", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.72, s=27.98, n=52, compute CI."}]}, "response": "95% CI ≈ [150.11, 165.33].", "meta": {"id": "WUA_STAT_25K_008227", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.57, s=34.74, n=169, compute CI."}]}, "response": "95% CI ≈ [167.33, 177.81].", "meta": {"id": "WUA_STAT_25K_008228", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.84, s=32.86, n=186, compute CI."}]}, "response": "95% CI ≈ [55.12, 64.56].", "meta": {"id": "WUA_STAT_25K_008229", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.72, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.172.", "meta": {"id": "WUA_STAT_25K_008230", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.81, μ=39.47, σ=17.41, compute z."}]}, "response": "z = 1.053.", "meta": {"id": "WUA_STAT_25K_008231", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.93, μ=57.99, σ=20.15, compute z."}]}, "response": "z = -1.591.", "meta": {"id": "WUA_STAT_25K_008232", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.81, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.688.", "meta": {"id": "WUA_STAT_25K_008233", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.34, μ=14.39, σ=1.02, compute z."}]}, "response": "z = 18.578.", "meta": {"id": "WUA_STAT_25K_008234", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.09, s=24.4, n=51, compute CI."}]}, "response": "95% CI ≈ [111.39, 124.79].", "meta": {"id": "WUA_STAT_25K_008235", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.067, sens=0.81, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.326.", "meta": {"id": "WUA_STAT_25K_008236", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.38, μ=69.43, σ=13.4, compute z."}]}, "response": "z = 1.638.", "meta": {"id": "WUA_STAT_25K_008237", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.82, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.432.", "meta": {"id": "WUA_STAT_25K_008238", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.67, compute E and Var."}]}, "response": "E[X]=7.37; Var[X]=2.432.", "meta": {"id": "WUA_STAT_25K_008239", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.14, μ=67.32, σ=14.32, compute z."}]}, "response": "z = -0.501.", "meta": {"id": "WUA_STAT_25K_008240", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.75, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.563.", "meta": {"id": "WUA_STAT_25K_008241", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.34, compute E and Var."}]}, "response": "E[X]=11.22; Var[X]=7.405.", "meta": {"id": "WUA_STAT_25K_008242", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.209, sens=0.78, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.433.", "meta": {"id": "WUA_STAT_25K_008243", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.016, sens=0.93, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.077.", "meta": {"id": "WUA_STAT_25K_008244", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=116.37, s=6.0, n=182, compute CI."}]}, "response": "95% CI ≈ [115.50, 117.24].", "meta": {"id": "WUA_STAT_25K_008245", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.81, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.488.", "meta": {"id": "WUA_STAT_25K_008246", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.54, μ=65.95, σ=13.96, compute z."}]}, "response": "z = 2.048.", "meta": {"id": "WUA_STAT_25K_008247", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.88, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.155.", "meta": {"id": "WUA_STAT_25K_008248", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.52, μ=8.87, σ=11.33, compute z."}]}, "response": "z = 0.234.", "meta": {"id": "WUA_STAT_25K_008249", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.67, μ=51.98, σ=21.8, compute z."}]}, "response": "z = -2.078.", "meta": {"id": "WUA_STAT_25K_008250", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.29, μ=28.19, σ=17.09, compute z."}]}, "response": "z = -0.813.", "meta": {"id": "WUA_STAT_25K_008251", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.108, sens=0.99, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.545.", "meta": {"id": "WUA_STAT_25K_008252", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.011, sens=0.83, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.071.", "meta": {"id": "WUA_STAT_25K_008253", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.67, μ=89.1, σ=8.51, compute z."}]}, "response": "z = -0.521.", "meta": {"id": "WUA_STAT_25K_008254", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.167, sens=0.95, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.514.", "meta": {"id": "WUA_STAT_25K_008255", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.78, compute E and Var."}]}, "response": "E[X]=14.82; Var[X]=3.260.", "meta": {"id": "WUA_STAT_25K_008256", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.169, sens=0.82, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.481.", "meta": {"id": "WUA_STAT_25K_008257", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.57, s=35.47, n=79, compute CI."}]}, "response": "95% CI ≈ [145.75, 161.39].", "meta": {"id": "WUA_STAT_25K_008258", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.071, sens=0.93, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.703.", "meta": {"id": "WUA_STAT_25K_008259", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.93, μ=48.77, σ=4.16, compute z."}]}, "response": "z = -10.779.", "meta": {"id": "WUA_STAT_25K_008260", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.09, compute E and Var."}]}, "response": "E[X]=1.89; Var[X]=1.720.", "meta": {"id": "WUA_STAT_25K_008261", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.17, s=10.17, n=135, compute CI."}]}, "response": "95% CI ≈ [73.45, 76.89].", "meta": {"id": "WUA_STAT_25K_008262", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.74, compute E and Var."}]}, "response": "E[X]=25.16; Var[X]=6.542.", "meta": {"id": "WUA_STAT_25K_008263", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.74, compute E and Var."}]}, "response": "E[X]=5.92; Var[X]=1.539.", "meta": {"id": "WUA_STAT_25K_008264", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.96, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.782.", "meta": {"id": "WUA_STAT_25K_008265", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.24, s=11.24, n=57, compute CI."}]}, "response": "95% CI ≈ [75.32, 81.16].", "meta": {"id": "WUA_STAT_25K_008266", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.18, compute E and Var."}]}, "response": "E[X]=1.44; Var[X]=1.181.", "meta": {"id": "WUA_STAT_25K_008267", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.33, μ=73.96, σ=22.31, compute z."}]}, "response": "z = -2.807.", "meta": {"id": "WUA_STAT_25K_008268", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.58, compute E and Var."}]}, "response": "E[X]=12.18; Var[X]=5.116.", "meta": {"id": "WUA_STAT_25K_008269", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.58, compute E and Var."}]}, "response": "E[X]=20.88; Var[X]=8.770.", "meta": {"id": "WUA_STAT_25K_008270", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.19, μ=5.77, σ=9.36, compute z."}]}, "response": "z = 4.318.", "meta": {"id": "WUA_STAT_25K_008271", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.28, compute E and Var."}]}, "response": "E[X]=10.36; Var[X]=7.459.", "meta": {"id": "WUA_STAT_25K_008272", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.33, compute E and Var."}]}, "response": "E[X]=4.29; Var[X]=2.874.", "meta": {"id": "WUA_STAT_25K_008273", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.48, μ=41.59, σ=16.03, compute z."}]}, "response": "z = 3.549.", "meta": {"id": "WUA_STAT_25K_008274", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.099, sens=0.78, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.271.", "meta": {"id": "WUA_STAT_25K_008275", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.69, compute E and Var."}]}, "response": "E[X]=22.77; Var[X]=7.059.", "meta": {"id": "WUA_STAT_25K_008276", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=96.14, s=19.7, n=81, compute CI."}]}, "response": "95% CI ≈ [91.85, 100.43].", "meta": {"id": "WUA_STAT_25K_008277", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.122, sens=0.86, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.428.", "meta": {"id": "WUA_STAT_25K_008278", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.42, s=36.27, n=135, compute CI."}]}, "response": "95% CI ≈ [93.30, 105.54].", "meta": {"id": "WUA_STAT_25K_008279", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.6, s=11.27, n=174, compute CI."}]}, "response": "95% CI ≈ [72.93, 76.27].", "meta": {"id": "WUA_STAT_25K_008280", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.24, sens=0.71, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.541.", "meta": {"id": "WUA_STAT_25K_008281", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.44, μ=35.38, σ=14.89, compute z."}]}, "response": "z = -1.473.", "meta": {"id": "WUA_STAT_25K_008282", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.23, compute E and Var."}]}, "response": "E[X]=2.53; Var[X]=1.948.", "meta": {"id": "WUA_STAT_25K_008283", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.058, sens=0.81, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.263.", "meta": {"id": "WUA_STAT_25K_008284", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.9, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.508.", "meta": {"id": "WUA_STAT_25K_008285", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.64, compute E and Var."}]}, "response": "E[X]=8.32; Var[X]=2.995.", "meta": {"id": "WUA_STAT_25K_008286", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.19, compute E and Var."}]}, "response": "E[X]=1.71; Var[X]=1.385.", "meta": {"id": "WUA_STAT_25K_008287", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.226, sens=0.72, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.478.", "meta": {"id": "WUA_STAT_25K_008288", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.28, s=11.07, n=50, compute CI."}]}, "response": "95% CI ≈ [98.21, 104.35].", "meta": {"id": "WUA_STAT_25K_008289", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.75, s=19.26, n=89, compute CI."}]}, "response": "95% CI ≈ [40.75, 48.75].", "meta": {"id": "WUA_STAT_25K_008290", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.72, compute E and Var."}]}, "response": "E[X]=26.64; Var[X]=7.459.", "meta": {"id": "WUA_STAT_25K_008291", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.06, compute E and Var."}]}, "response": "E[X]=1.08; Var[X]=1.015.", "meta": {"id": "WUA_STAT_25K_008292", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.072, sens=0.86, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.690.", "meta": {"id": "WUA_STAT_25K_008293", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.62, compute E and Var."}]}, "response": "E[X]=18.60; Var[X]=7.068.", "meta": {"id": "WUA_STAT_25K_008294", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.28, s=36.03, n=37, compute CI."}]}, "response": "95% CI ≈ [76.67, 99.89].", "meta": {"id": "WUA_STAT_25K_008295", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.93, μ=81.77, σ=21.0, compute z."}]}, "response": "z = -1.754.", "meta": {"id": "WUA_STAT_25K_008296", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.32, μ=40.1, σ=21.77, compute z."}]}, "response": "z = -1.460.", "meta": {"id": "WUA_STAT_25K_008297", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.78, μ=88.01, σ=9.92, compute z."}]}, "response": "z = 0.279.", "meta": {"id": "WUA_STAT_25K_008298", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.123, sens=0.98, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.604.", "meta": {"id": "WUA_STAT_25K_008299", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.095, sens=0.83, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.685.", "meta": {"id": "WUA_STAT_25K_008300", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.06, s=21.9, n=128, compute CI."}]}, "response": "95% CI ≈ [116.27, 123.85].", "meta": {"id": "WUA_STAT_25K_008301", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.14, compute E and Var."}]}, "response": "E[X]=3.92; Var[X]=3.371.", "meta": {"id": "WUA_STAT_25K_008302", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.75, μ=82.28, σ=13.0, compute z."}]}, "response": "z = 0.728.", "meta": {"id": "WUA_STAT_25K_008303", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.86, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.644.", "meta": {"id": "WUA_STAT_25K_008304", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.024, sens=0.84, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.090.", "meta": {"id": "WUA_STAT_25K_008305", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.29, s=28.36, n=41, compute CI."}]}, "response": "95% CI ≈ [70.61, 87.97].", "meta": {"id": "WUA_STAT_25K_008306", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.89, s=2.92, n=190, compute CI."}]}, "response": "95% CI ≈ [177.47, 178.31].", "meta": {"id": "WUA_STAT_25K_008307", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.91, μ=33.57, σ=19.13, compute z."}]}, "response": "z = -0.662.", "meta": {"id": "WUA_STAT_25K_008308", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.47, compute E and Var."}]}, "response": "E[X]=18.80; Var[X]=9.964.", "meta": {"id": "WUA_STAT_25K_008309", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.81, μ=13.08, σ=13.91, compute z."}]}, "response": "z = 4.869.", "meta": {"id": "WUA_STAT_25K_008310", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.93, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.296.", "meta": {"id": "WUA_STAT_25K_008311", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.81, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.791.", "meta": {"id": "WUA_STAT_25K_008312", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.43, s=32.24, n=70, compute CI."}]}, "response": "95% CI ≈ [110.88, 125.98].", "meta": {"id": "WUA_STAT_25K_008313", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.43, compute E and Var."}]}, "response": "E[X]=15.48; Var[X]=8.824.", "meta": {"id": "WUA_STAT_25K_008314", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.48, μ=24.46, σ=9.58, compute z."}]}, "response": "z = 3.969.", "meta": {"id": "WUA_STAT_25K_008315", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.98, μ=88.21, σ=22.1, compute z."}]}, "response": "z = 0.352.", "meta": {"id": "WUA_STAT_25K_008316", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.23, s=15.48, n=186, compute CI."}]}, "response": "95% CI ≈ [63.01, 67.45].", "meta": {"id": "WUA_STAT_25K_008317", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.73, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.326.", "meta": {"id": "WUA_STAT_25K_008318", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.87, μ=40.06, σ=9.55, compute z."}]}, "response": "z = -0.858.", "meta": {"id": "WUA_STAT_25K_008319", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.3, compute E and Var."}]}, "response": "E[X]=10.80; Var[X]=7.560.", "meta": {"id": "WUA_STAT_25K_008320", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.16, compute E and Var."}]}, "response": "E[X]=1.76; Var[X]=1.478.", "meta": {"id": "WUA_STAT_25K_008321", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.79, compute E and Var."}]}, "response": "E[X]=30.81; Var[X]=6.470.", "meta": {"id": "WUA_STAT_25K_008322", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.04, μ=34.04, σ=18.0, compute z."}]}, "response": "z = 0.500.", "meta": {"id": "WUA_STAT_25K_008323", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.29, s=38.18, n=73, compute CI."}]}, "response": "95% CI ≈ [79.53, 97.05].", "meta": {"id": "WUA_STAT_25K_008324", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=147.49, s=21.68, n=186, compute CI."}]}, "response": "95% CI ≈ [144.37, 150.61].", "meta": {"id": "WUA_STAT_25K_008325", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=162.42, s=26.97, n=75, compute CI."}]}, "response": "95% CI ≈ [156.32, 168.52].", "meta": {"id": "WUA_STAT_25K_008326", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.35, μ=95.8, σ=15.5, compute z."}]}, "response": "z = -0.674.", "meta": {"id": "WUA_STAT_25K_008327", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.89, μ=76.66, σ=24.75, compute z."}]}, "response": "z = -2.657.", "meta": {"id": "WUA_STAT_25K_008328", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.87, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.733.", "meta": {"id": "WUA_STAT_25K_008329", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.131, sens=0.85, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.461.", "meta": {"id": "WUA_STAT_25K_008330", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.64, μ=47.55, σ=20.73, compute z."}]}, "response": "z = -1.539.", "meta": {"id": "WUA_STAT_25K_008331", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.77, compute E and Var."}]}, "response": "E[X]=6.93; Var[X]=1.594.", "meta": {"id": "WUA_STAT_25K_008332", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.94, s=30.58, n=88, compute CI."}]}, "response": "95% CI ≈ [24.55, 37.33].", "meta": {"id": "WUA_STAT_25K_008333", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.55, μ=45.45, σ=16.71, compute z."}]}, "response": "z = 0.963.", "meta": {"id": "WUA_STAT_25K_008334", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.28, μ=37.36, σ=21.66, compute z."}]}, "response": "z = -0.004.", "meta": {"id": "WUA_STAT_25K_008335", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.223, sens=0.74, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.620.", "meta": {"id": "WUA_STAT_25K_008336", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.99, s=21.85, n=95, compute CI."}]}, "response": "95% CI ≈ [30.60, 39.38].", "meta": {"id": "WUA_STAT_25K_008337", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.8, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.732.", "meta": {"id": "WUA_STAT_25K_008338", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.74, μ=93.3, σ=4.01, compute z."}]}, "response": "z = -6.125.", "meta": {"id": "WUA_STAT_25K_008339", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.57, compute E and Var."}]}, "response": "E[X]=17.10; Var[X]=7.353.", "meta": {"id": "WUA_STAT_25K_008340", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.153, sens=0.96, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.374.", "meta": {"id": "WUA_STAT_25K_008341", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.29, s=35.79, n=87, compute CI."}]}, "response": "95% CI ≈ [62.77, 77.81].", "meta": {"id": "WUA_STAT_25K_008342", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.86, s=37.04, n=102, compute CI."}]}, "response": "95% CI ≈ [180.67, 195.05].", "meta": {"id": "WUA_STAT_25K_008343", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.34, compute E and Var."}]}, "response": "E[X]=9.18; Var[X]=6.059.", "meta": {"id": "WUA_STAT_25K_008344", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.64, μ=35.97, σ=3.73, compute z."}]}, "response": "z = 7.150.", "meta": {"id": "WUA_STAT_25K_008345", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.54, s=24.13, n=153, compute CI."}]}, "response": "95% CI ≈ [70.72, 78.36].", "meta": {"id": "WUA_STAT_25K_008346", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.13, compute E and Var."}]}, "response": "E[X]=3.90; Var[X]=3.393.", "meta": {"id": "WUA_STAT_25K_008347", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=71.85, s=33.15, n=168, compute CI."}]}, "response": "95% CI ≈ [66.84, 76.86].", "meta": {"id": "WUA_STAT_25K_008348", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.24, s=5.66, n=33, compute CI."}]}, "response": "95% CI ≈ [149.31, 153.17].", "meta": {"id": "WUA_STAT_25K_008349", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.141, sens=0.86, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.361.", "meta": {"id": "WUA_STAT_25K_008350", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.162, sens=0.88, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.448.", "meta": {"id": "WUA_STAT_25K_008351", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.76, compute E and Var."}]}, "response": "E[X]=29.64; Var[X]=7.114.", "meta": {"id": "WUA_STAT_25K_008352", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.89, s=37.13, n=141, compute CI."}]}, "response": "95% CI ≈ [145.76, 158.02].", "meta": {"id": "WUA_STAT_25K_008353", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.96, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.459.", "meta": {"id": "WUA_STAT_25K_008354", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.56, μ=72.67, σ=13.69, compute z."}]}, "response": "z = -2.857.", "meta": {"id": "WUA_STAT_25K_008355", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.223, sens=0.85, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.635.", "meta": {"id": "WUA_STAT_25K_008356", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.81, μ=52.6, σ=20.11, compute z."}]}, "response": "z = 0.607.", "meta": {"id": "WUA_STAT_25K_008357", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.6, compute E and Var."}]}, "response": "E[X]=12.00; Var[X]=4.800.", "meta": {"id": "WUA_STAT_25K_008358", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.01, sens=0.84, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.034.", "meta": {"id": "WUA_STAT_25K_008359", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.193, sens=0.81, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.492.", "meta": {"id": "WUA_STAT_25K_008360", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.2, compute E and Var."}]}, "response": "E[X]=2.20; Var[X]=1.760.", "meta": {"id": "WUA_STAT_25K_008361", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.224, sens=0.7, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.692.", "meta": {"id": "WUA_STAT_25K_008362", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.36, compute E and Var."}]}, "response": "E[X]=10.44; Var[X]=6.682.", "meta": {"id": "WUA_STAT_25K_008363", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.72, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.241.", "meta": {"id": "WUA_STAT_25K_008364", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.93, μ=85.9, σ=3.91, compute z."}]}, "response": "z = -4.596.", "meta": {"id": "WUA_STAT_25K_008365", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.17, compute E and Var."}]}, "response": "E[X]=6.46; Var[X]=5.362.", "meta": {"id": "WUA_STAT_25K_008366", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.78, compute E and Var."}]}, "response": "E[X]=28.08; Var[X]=6.178.", "meta": {"id": "WUA_STAT_25K_008367", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.94, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.433.", "meta": {"id": "WUA_STAT_25K_008368", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.96, μ=77.61, σ=12.36, compute z."}]}, "response": "z = -2.722.", "meta": {"id": "WUA_STAT_25K_008369", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.94, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.697.", "meta": {"id": "WUA_STAT_25K_008370", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.81, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.421.", "meta": {"id": "WUA_STAT_25K_008371", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.2, μ=15.8, σ=21.54, compute z."}]}, "response": "z = 2.108.", "meta": {"id": "WUA_STAT_25K_008372", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.32, μ=61.4, σ=12.08, compute z."}]}, "response": "z = -2.738.", "meta": {"id": "WUA_STAT_25K_008373", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.41, s=33.31, n=120, compute CI."}]}, "response": "95% CI ≈ [108.45, 120.37].", "meta": {"id": "WUA_STAT_25K_008374", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.203, sens=0.87, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.613.", "meta": {"id": "WUA_STAT_25K_008375", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.17, sens=0.75, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.434.", "meta": {"id": "WUA_STAT_25K_008376", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.71, compute E and Var."}]}, "response": "E[X]=26.98; Var[X]=7.824.", "meta": {"id": "WUA_STAT_25K_008377", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.187, sens=0.84, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.659.", "meta": {"id": "WUA_STAT_25K_008378", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.32, μ=0.11, σ=4.58, compute z."}]}, "response": "z = 19.478.", "meta": {"id": "WUA_STAT_25K_008379", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.21, sens=0.74, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.641.", "meta": {"id": "WUA_STAT_25K_008380", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.22, s=24.19, n=124, compute CI."}]}, "response": "95% CI ≈ [49.96, 58.48].", "meta": {"id": "WUA_STAT_25K_008381", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.01, μ=28.75, σ=12.09, compute z."}]}, "response": "z = -0.806.", "meta": {"id": "WUA_STAT_25K_008382", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.75, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.274.", "meta": {"id": "WUA_STAT_25K_008383", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.92, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.624.", "meta": {"id": "WUA_STAT_25K_008384", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.041, sens=0.75, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.517.", "meta": {"id": "WUA_STAT_25K_008385", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.73, μ=65.38, σ=20.58, compute z."}]}, "response": "z = -0.906.", "meta": {"id": "WUA_STAT_25K_008386", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.42, compute E and Var."}]}, "response": "E[X]=8.82; Var[X]=5.116.", "meta": {"id": "WUA_STAT_25K_008387", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=95.28, s=34.07, n=116, compute CI."}]}, "response": "95% CI ≈ [89.08, 101.48].", "meta": {"id": "WUA_STAT_25K_008388", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.67, μ=6.2, σ=2.78, compute z."}]}, "response": "z = 21.032.", "meta": {"id": "WUA_STAT_25K_008389", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.244, sens=0.85, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.846.", "meta": {"id": "WUA_STAT_25K_008390", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.93, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.701.", "meta": {"id": "WUA_STAT_25K_008391", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.7, compute E and Var."}]}, "response": "E[X]=3.50; Var[X]=1.050.", "meta": {"id": "WUA_STAT_25K_008392", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.78, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.053.", "meta": {"id": "WUA_STAT_25K_008393", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.74, compute E and Var."}]}, "response": "E[X]=14.80; Var[X]=3.848.", "meta": {"id": "WUA_STAT_25K_008394", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.29, μ=41.15, σ=24.72, compute z."}]}, "response": "z = -0.075.", "meta": {"id": "WUA_STAT_25K_008395", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.84, μ=1.72, σ=9.52, compute z."}]}, "response": "z = 6.105.", "meta": {"id": "WUA_STAT_25K_008396", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.1, compute E and Var."}]}, "response": "E[X]=0.90; Var[X]=0.810.", "meta": {"id": "WUA_STAT_25K_008397", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.124, sens=0.76, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.544.", "meta": {"id": "WUA_STAT_25K_008398", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.4, s=29.82, n=104, compute CI."}]}, "response": "95% CI ≈ [7.67, 19.13].", "meta": {"id": "WUA_STAT_25K_008399", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.9, μ=77.06, σ=2.6, compute z."}]}, "response": "z = -7.754.", "meta": {"id": "WUA_STAT_25K_008400", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.197, sens=0.79, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.795.", "meta": {"id": "WUA_STAT_25K_008401", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.7, compute E and Var."}]}, "response": "E[X]=16.10; Var[X]=4.830.", "meta": {"id": "WUA_STAT_25K_008402", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.0, μ=39.9, σ=1.24, compute z."}]}, "response": "z = 36.371.", "meta": {"id": "WUA_STAT_25K_008403", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.15, μ=54.04, σ=15.05, compute z."}]}, "response": "z = 2.266.", "meta": {"id": "WUA_STAT_25K_008404", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.71, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.363.", "meta": {"id": "WUA_STAT_25K_008405", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.89, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.774.", "meta": {"id": "WUA_STAT_25K_008406", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.168, sens=0.7, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.352.", "meta": {"id": "WUA_STAT_25K_008407", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.29, compute E and Var."}]}, "response": "E[X]=3.19; Var[X]=2.265.", "meta": {"id": "WUA_STAT_25K_008408", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.45, s=7.49, n=104, compute CI."}]}, "response": "95% CI ≈ [11.01, 13.89].", "meta": {"id": "WUA_STAT_25K_008409", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.11, compute E and Var."}]}, "response": "E[X]=3.30; Var[X]=2.937.", "meta": {"id": "WUA_STAT_25K_008410", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.79, compute E and Var."}]}, "response": "E[X]=20.54; Var[X]=4.313.", "meta": {"id": "WUA_STAT_25K_008411", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.69, μ=80.0, σ=9.78, compute z."}]}, "response": "z = -3.304.", "meta": {"id": "WUA_STAT_25K_008412", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.231, sens=0.91, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.503.", "meta": {"id": "WUA_STAT_25K_008413", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.99, μ=45.18, σ=5.82, compute z."}]}, "response": "z = 0.311.", "meta": {"id": "WUA_STAT_25K_008414", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.52, μ=28.24, σ=21.84, compute z."}]}, "response": "z = -0.308.", "meta": {"id": "WUA_STAT_25K_008415", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.118, sens=0.77, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.720.", "meta": {"id": "WUA_STAT_25K_008416", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.39, compute E and Var."}]}, "response": "E[X]=12.09; Var[X]=7.375.", "meta": {"id": "WUA_STAT_25K_008417", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.75, compute E and Var."}]}, "response": "E[X]=7.50; Var[X]=1.875.", "meta": {"id": "WUA_STAT_25K_008418", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.93, μ=60.76, σ=18.22, compute z."}]}, "response": "z = -0.265.", "meta": {"id": "WUA_STAT_25K_008419", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.062, sens=0.92, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.670.", "meta": {"id": "WUA_STAT_25K_008420", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.78, s=17.82, n=179, compute CI."}]}, "response": "95% CI ≈ [133.17, 138.39].", "meta": {"id": "WUA_STAT_25K_008421", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.38, s=31.06, n=63, compute CI."}]}, "response": "95% CI ≈ [186.71, 202.05].", "meta": {"id": "WUA_STAT_25K_008422", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.0, μ=65.58, σ=18.14, compute z."}]}, "response": "z = -2.292.", "meta": {"id": "WUA_STAT_25K_008423", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.53, s=2.72, n=200, compute CI."}]}, "response": "95% CI ≈ [194.15, 194.91].", "meta": {"id": "WUA_STAT_25K_008424", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.71, μ=80.83, σ=22.88, compute z."}]}, "response": "z = -2.234.", "meta": {"id": "WUA_STAT_25K_008425", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.197, sens=0.92, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.958.", "meta": {"id": "WUA_STAT_25K_008426", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.17, compute E and Var."}]}, "response": "E[X]=6.63; Var[X]=5.503.", "meta": {"id": "WUA_STAT_25K_008427", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.37, s=5.23, n=122, compute CI."}]}, "response": "95% CI ≈ [112.44, 114.30].", "meta": {"id": "WUA_STAT_25K_008428", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.99, s=1.33, n=142, compute CI."}]}, "response": "95% CI ≈ [88.77, 89.21].", "meta": {"id": "WUA_STAT_25K_008429", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.06, s=4.56, n=144, compute CI."}]}, "response": "95% CI ≈ [151.32, 152.80].", "meta": {"id": "WUA_STAT_25K_008430", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.4, μ=61.54, σ=11.66, compute z."}]}, "response": "z = -4.214.", "meta": {"id": "WUA_STAT_25K_008431", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.89, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.360.", "meta": {"id": "WUA_STAT_25K_008432", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.38, μ=65.53, σ=16.39, compute z."}]}, "response": "z = -3.792.", "meta": {"id": "WUA_STAT_25K_008433", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.55, μ=44.84, σ=10.75, compute z."}]}, "response": "z = 1.089.", "meta": {"id": "WUA_STAT_25K_008434", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.67, μ=29.38, σ=4.89, compute z."}]}, "response": "z = 4.763.", "meta": {"id": "WUA_STAT_25K_008435", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.75, compute E and Var."}]}, "response": "E[X]=14.25; Var[X]=3.562.", "meta": {"id": "WUA_STAT_25K_008436", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.99, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.490.", "meta": {"id": "WUA_STAT_25K_008437", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.59, μ=81.88, σ=13.79, compute z."}]}, "response": "z = -4.227.", "meta": {"id": "WUA_STAT_25K_008438", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.51, μ=21.95, σ=4.57, compute z."}]}, "response": "z = 4.499.", "meta": {"id": "WUA_STAT_25K_008439", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.77, s=11.92, n=57, compute CI."}]}, "response": "95% CI ≈ [54.68, 60.86].", "meta": {"id": "WUA_STAT_25K_008440", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.83, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.370.", "meta": {"id": "WUA_STAT_25K_008441", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.68, s=35.4, n=166, compute CI."}]}, "response": "95% CI ≈ [179.29, 190.07].", "meta": {"id": "WUA_STAT_25K_008442", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.0, s=3.26, n=92, compute CI."}]}, "response": "95% CI ≈ [121.33, 122.67].", "meta": {"id": "WUA_STAT_25K_008443", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.091, sens=0.9, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.291.", "meta": {"id": "WUA_STAT_25K_008444", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.41, s=18.83, n=151, compute CI."}]}, "response": "95% CI ≈ [32.41, 38.41].", "meta": {"id": "WUA_STAT_25K_008445", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.96, μ=7.54, σ=23.3, compute z."}]}, "response": "z = 1.649.", "meta": {"id": "WUA_STAT_25K_008446", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.46, compute E and Var."}]}, "response": "E[X]=4.60; Var[X]=2.484.", "meta": {"id": "WUA_STAT_25K_008447", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.71, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.288.", "meta": {"id": "WUA_STAT_25K_008448", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.74, compute E and Var."}]}, "response": "E[X]=25.16; Var[X]=6.542.", "meta": {"id": "WUA_STAT_25K_008449", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.162, sens=0.76, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.361.", "meta": {"id": "WUA_STAT_25K_008450", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.06, μ=86.13, σ=24.13, compute z."}]}, "response": "z = -3.360.", "meta": {"id": "WUA_STAT_25K_008451", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.42, μ=58.15, σ=19.98, compute z."}]}, "response": "z = -0.487.", "meta": {"id": "WUA_STAT_25K_008452", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.8, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.449.", "meta": {"id": "WUA_STAT_25K_008453", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.07, compute E and Var."}]}, "response": "E[X]=1.33; Var[X]=1.237.", "meta": {"id": "WUA_STAT_25K_008454", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.94, μ=24.56, σ=21.95, compute z."}]}, "response": "z = 0.154.", "meta": {"id": "WUA_STAT_25K_008455", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.205, sens=0.78, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.528.", "meta": {"id": "WUA_STAT_25K_008456", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.62, compute E and Var."}]}, "response": "E[X]=24.18; Var[X]=9.188.", "meta": {"id": "WUA_STAT_25K_008457", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.85, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.421.", "meta": {"id": "WUA_STAT_25K_008458", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.26, s=20.21, n=18, compute CI."}]}, "response": "95% CI ≈ [176.92, 195.60].", "meta": {"id": "WUA_STAT_25K_008459", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.29, compute E and Var."}]}, "response": "E[X]=11.60; Var[X]=8.236.", "meta": {"id": "WUA_STAT_25K_008460", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.71, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.562.", "meta": {"id": "WUA_STAT_25K_008461", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.148, sens=0.97, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.367.", "meta": {"id": "WUA_STAT_25K_008462", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.88, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.533.", "meta": {"id": "WUA_STAT_25K_008463", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.37, s=24.36, n=120, compute CI."}]}, "response": "95% CI ≈ [8.01, 16.73].", "meta": {"id": "WUA_STAT_25K_008464", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.37, μ=19.39, σ=5.84, compute z."}]}, "response": "z = 2.051.", "meta": {"id": "WUA_STAT_25K_008465", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.1, compute E and Var."}]}, "response": "E[X]=1.90; Var[X]=1.710.", "meta": {"id": "WUA_STAT_25K_008466", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=196.08, s=28.25, n=194, compute CI."}]}, "response": "95% CI ≈ [192.10, 200.06].", "meta": {"id": "WUA_STAT_25K_008467", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.16, compute E and Var."}]}, "response": "E[X]=5.44; Var[X]=4.570.", "meta": {"id": "WUA_STAT_25K_008468", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.0, μ=48.97, σ=2.78, compute z."}]}, "response": "z = -2.147.", "meta": {"id": "WUA_STAT_25K_008469", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.95, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.439.", "meta": {"id": "WUA_STAT_25K_008470", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.014, sens=0.83, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.078.", "meta": {"id": "WUA_STAT_25K_008471", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.49, compute E and Var."}]}, "response": "E[X]=16.66; Var[X]=8.497.", "meta": {"id": "WUA_STAT_25K_008472", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.19, compute E and Var."}]}, "response": "E[X]=3.23; Var[X]=2.616.", "meta": {"id": "WUA_STAT_25K_008473", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.6, μ=43.44, σ=8.85, compute z."}]}, "response": "z = 4.086.", "meta": {"id": "WUA_STAT_25K_008474", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.01, s=29.11, n=85, compute CI."}]}, "response": "95% CI ≈ [105.82, 118.20].", "meta": {"id": "WUA_STAT_25K_008475", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.38, compute E and Var."}]}, "response": "E[X]=13.68; Var[X]=8.482.", "meta": {"id": "WUA_STAT_25K_008476", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.76, compute E and Var."}]}, "response": "E[X]=6.84; Var[X]=1.642.", "meta": {"id": "WUA_STAT_25K_008477", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.95, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.910.", "meta": {"id": "WUA_STAT_25K_008478", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.07, compute E and Var."}]}, "response": "E[X]=2.66; Var[X]=2.474.", "meta": {"id": "WUA_STAT_25K_008479", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.52, μ=64.38, σ=18.82, compute z."}]}, "response": "z = -1.640.", "meta": {"id": "WUA_STAT_25K_008480", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.133, sens=0.84, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.539.", "meta": {"id": "WUA_STAT_25K_008481", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.29, compute E and Var."}]}, "response": "E[X]=9.57; Var[X]=6.795.", "meta": {"id": "WUA_STAT_25K_008482", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.246, sens=0.88, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.578.", "meta": {"id": "WUA_STAT_25K_008483", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.78, μ=4.43, σ=1.04, compute z."}]}, "response": "z = 13.798.", "meta": {"id": "WUA_STAT_25K_008484", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.118, sens=0.92, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.528.", "meta": {"id": "WUA_STAT_25K_008485", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.11, compute E and Var."}]}, "response": "E[X]=4.07; Var[X]=3.622.", "meta": {"id": "WUA_STAT_25K_008486", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.84, s=7.76, n=52, compute CI."}]}, "response": "95% CI ≈ [58.73, 62.95].", "meta": {"id": "WUA_STAT_25K_008487", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.244, sens=0.86, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.716.", "meta": {"id": "WUA_STAT_25K_008488", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.58, compute E and Var."}]}, "response": "E[X]=2.90; Var[X]=1.218.", "meta": {"id": "WUA_STAT_25K_008489", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.62, s=39.49, n=145, compute CI."}]}, "response": "95% CI ≈ [125.19, 138.05].", "meta": {"id": "WUA_STAT_25K_008490", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.0, μ=92.83, σ=4.12, compute z."}]}, "response": "z = -6.998.", "meta": {"id": "WUA_STAT_25K_008491", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.18, compute E and Var."}]}, "response": "E[X]=4.32; Var[X]=3.542.", "meta": {"id": "WUA_STAT_25K_008492", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.61, compute E and Var."}]}, "response": "E[X]=16.47; Var[X]=6.423.", "meta": {"id": "WUA_STAT_25K_008493", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=168.66, s=21.67, n=188, compute CI."}]}, "response": "95% CI ≈ [165.56, 171.76].", "meta": {"id": "WUA_STAT_25K_008494", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.51, compute E and Var."}]}, "response": "E[X]=12.75; Var[X]=6.247.", "meta": {"id": "WUA_STAT_25K_008495", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.81, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.484.", "meta": {"id": "WUA_STAT_25K_008496", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.83, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.573.", "meta": {"id": "WUA_STAT_25K_008497", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.203, sens=0.89, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.476.", "meta": {"id": "WUA_STAT_25K_008498", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.81, s=16.05, n=171, compute CI."}]}, "response": "95% CI ≈ [144.40, 149.22].", "meta": {"id": "WUA_STAT_25K_008499", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.94, μ=31.67, σ=13.43, compute z."}]}, "response": "z = 1.286.", "meta": {"id": "WUA_STAT_25K_008500", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.12, s=30.02, n=19, compute CI."}]}, "response": "95% CI ≈ [117.62, 144.62].", "meta": {"id": "WUA_STAT_25K_008501", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.029, sens=0.89, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.347.", "meta": {"id": "WUA_STAT_25K_008502", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.157, sens=0.75, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.368.", "meta": {"id": "WUA_STAT_25K_008503", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.149, sens=0.95, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.806.", "meta": {"id": "WUA_STAT_25K_008504", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.89, s=23.71, n=144, compute CI."}]}, "response": "95% CI ≈ [190.02, 197.76].", "meta": {"id": "WUA_STAT_25K_008505", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.52, s=12.3, n=76, compute CI."}]}, "response": "95% CI ≈ [117.75, 123.29].", "meta": {"id": "WUA_STAT_25K_008506", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.25, μ=86.97, σ=2.93, compute z."}]}, "response": "z = -8.096.", "meta": {"id": "WUA_STAT_25K_008507", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=116.73, s=2.54, n=165, compute CI."}]}, "response": "95% CI ≈ [116.34, 117.12].", "meta": {"id": "WUA_STAT_25K_008508", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.032, sens=0.75, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.090.", "meta": {"id": "WUA_STAT_25K_008509", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.58, s=36.45, n=33, compute CI."}]}, "response": "95% CI ≈ [101.14, 126.02].", "meta": {"id": "WUA_STAT_25K_008510", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.78, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.035.", "meta": {"id": "WUA_STAT_25K_008511", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.04, s=3.7, n=198, compute CI."}]}, "response": "95% CI ≈ [78.52, 79.56].", "meta": {"id": "WUA_STAT_25K_008512", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.88, μ=46.25, σ=2.31, compute z."}]}, "response": "z = -4.056.", "meta": {"id": "WUA_STAT_25K_008513", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.77, compute E and Var."}]}, "response": "E[X]=23.10; Var[X]=5.313.", "meta": {"id": "WUA_STAT_25K_008514", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.33, compute E and Var."}]}, "response": "E[X]=8.91; Var[X]=5.970.", "meta": {"id": "WUA_STAT_25K_008515", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.39, compute E and Var."}]}, "response": "E[X]=14.04; Var[X]=8.564.", "meta": {"id": "WUA_STAT_25K_008516", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.63, μ=34.42, σ=21.4, compute z."}]}, "response": "z = 2.113.", "meta": {"id": "WUA_STAT_25K_008517", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.73, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.436.", "meta": {"id": "WUA_STAT_25K_008518", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.67, compute E and Var."}]}, "response": "E[X]=12.73; Var[X]=4.201.", "meta": {"id": "WUA_STAT_25K_008519", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.01, sens=0.72, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.046.", "meta": {"id": "WUA_STAT_25K_008520", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.93, μ=24.51, σ=13.58, compute z."}]}, "response": "z = 3.124.", "meta": {"id": "WUA_STAT_25K_008521", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.72, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.482.", "meta": {"id": "WUA_STAT_25K_008522", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.43, s=27.04, n=70, compute CI."}]}, "response": "95% CI ≈ [154.10, 166.76].", "meta": {"id": "WUA_STAT_25K_008523", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.16, compute E and Var."}]}, "response": "E[X]=2.88; Var[X]=2.419.", "meta": {"id": "WUA_STAT_25K_008524", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=169.42, s=9.67, n=199, compute CI."}]}, "response": "95% CI ≈ [168.08, 170.76].", "meta": {"id": "WUA_STAT_25K_008525", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.65, s=4.78, n=38, compute CI."}]}, "response": "95% CI ≈ [139.13, 142.17].", "meta": {"id": "WUA_STAT_25K_008526", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.75, μ=54.16, σ=6.76, compute z."}]}, "response": "z = -1.540.", "meta": {"id": "WUA_STAT_25K_008527", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.033, sens=0.87, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.102.", "meta": {"id": "WUA_STAT_25K_008528", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.39, compute E and Var."}]}, "response": "E[X]=15.21; Var[X]=9.278.", "meta": {"id": "WUA_STAT_25K_008529", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.7, μ=72.25, σ=1.55, compute z."}]}, "response": "z = 17.710.", "meta": {"id": "WUA_STAT_25K_008530", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.45, s=38.63, n=147, compute CI."}]}, "response": "95% CI ≈ [121.21, 133.69].", "meta": {"id": "WUA_STAT_25K_008531", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.228, sens=0.96, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.625.", "meta": {"id": "WUA_STAT_25K_008532", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.19, s=21.64, n=60, compute CI."}]}, "response": "95% CI ≈ [136.71, 147.67].", "meta": {"id": "WUA_STAT_25K_008533", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.74, compute E and Var."}]}, "response": "E[X]=5.92; Var[X]=1.539.", "meta": {"id": "WUA_STAT_25K_008534", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=26.64, s=28.07, n=48, compute CI."}]}, "response": "95% CI ≈ [18.70, 34.58].", "meta": {"id": "WUA_STAT_25K_008535", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.7, compute E and Var."}]}, "response": "E[X]=4.90; Var[X]=1.470.", "meta": {"id": "WUA_STAT_25K_008536", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=126.71, s=29.38, n=63, compute CI."}]}, "response": "95% CI ≈ [119.45, 133.97].", "meta": {"id": "WUA_STAT_25K_008537", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.45, compute E and Var."}]}, "response": "E[X]=10.35; Var[X]=5.692.", "meta": {"id": "WUA_STAT_25K_008538", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.88, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.673.", "meta": {"id": "WUA_STAT_25K_008539", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.32, μ=74.68, σ=11.94, compute z."}]}, "response": "z = -1.789.", "meta": {"id": "WUA_STAT_25K_008540", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.52, compute E and Var."}]}, "response": "E[X]=11.96; Var[X]=5.741.", "meta": {"id": "WUA_STAT_25K_008541", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.94, μ=65.88, σ=5.72, compute z."}]}, "response": "z = 3.157.", "meta": {"id": "WUA_STAT_25K_008542", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.59, s=38.81, n=89, compute CI."}]}, "response": "95% CI ≈ [37.53, 53.65].", "meta": {"id": "WUA_STAT_25K_008543", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.026, sens=0.92, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.380.", "meta": {"id": "WUA_STAT_25K_008544", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.16, compute E and Var."}]}, "response": "E[X]=5.92; Var[X]=4.973.", "meta": {"id": "WUA_STAT_25K_008545", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.69, μ=22.47, σ=22.4, compute z."}]}, "response": "z = 1.349.", "meta": {"id": "WUA_STAT_25K_008546", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.2, s=4.38, n=102, compute CI."}]}, "response": "95% CI ≈ [118.35, 120.05].", "meta": {"id": "WUA_STAT_25K_008547", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.97, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.501.", "meta": {"id": "WUA_STAT_25K_008548", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.057, sens=0.89, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.402.", "meta": {"id": "WUA_STAT_25K_008549", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.34, μ=39.55, σ=15.86, compute z."}]}, "response": "z = 1.689.", "meta": {"id": "WUA_STAT_25K_008550", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.44, μ=21.16, σ=23.68, compute z."}]}, "response": "z = -0.495.", "meta": {"id": "WUA_STAT_25K_008551", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.28, compute E and Var."}]}, "response": "E[X]=9.52; Var[X]=6.854.", "meta": {"id": "WUA_STAT_25K_008552", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.093, sens=0.91, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.459.", "meta": {"id": "WUA_STAT_25K_008553", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.018, sens=0.74, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.046.", "meta": {"id": "WUA_STAT_25K_008554", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.21, compute E and Var."}]}, "response": "E[X]=5.25; Var[X]=4.147.", "meta": {"id": "WUA_STAT_25K_008555", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.96, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.773.", "meta": {"id": "WUA_STAT_25K_008556", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.53, μ=32.37, σ=3.76, compute z."}]}, "response": "z = 8.287.", "meta": {"id": "WUA_STAT_25K_008557", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.7, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=1.680.", "meta": {"id": "WUA_STAT_25K_008558", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.64, μ=57.96, σ=23.3, compute z."}]}, "response": "z = -1.130.", "meta": {"id": "WUA_STAT_25K_008559", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.58, s=37.74, n=158, compute CI."}]}, "response": "95% CI ≈ [74.70, 86.46].", "meta": {"id": "WUA_STAT_25K_008560", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.66, compute E and Var."}]}, "response": "E[X]=15.18; Var[X]=5.161.", "meta": {"id": "WUA_STAT_25K_008561", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.79, compute E and Var."}]}, "response": "E[X]=30.02; Var[X]=6.304.", "meta": {"id": "WUA_STAT_25K_008562", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.103, sens=0.91, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.511.", "meta": {"id": "WUA_STAT_25K_008563", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.9, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.346.", "meta": {"id": "WUA_STAT_25K_008564", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.58, μ=61.42, σ=5.79, compute z."}]}, "response": "z = -5.845.", "meta": {"id": "WUA_STAT_25K_008565", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.053, sens=0.9, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.265.", "meta": {"id": "WUA_STAT_25K_008566", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.63, s=32.56, n=139, compute CI."}]}, "response": "95% CI ≈ [188.22, 199.04].", "meta": {"id": "WUA_STAT_25K_008567", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.65, μ=15.75, σ=14.76, compute z."}]}, "response": "z = 4.533.", "meta": {"id": "WUA_STAT_25K_008568", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.53, compute E and Var."}]}, "response": "E[X]=20.67; Var[X]=9.715.", "meta": {"id": "WUA_STAT_25K_008569", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.23, compute E and Var."}]}, "response": "E[X]=2.30; Var[X]=1.771.", "meta": {"id": "WUA_STAT_25K_008570", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.17, compute E and Var."}]}, "response": "E[X]=1.19; Var[X]=0.988.", "meta": {"id": "WUA_STAT_25K_008571", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.45, compute E and Var."}]}, "response": "E[X]=13.50; Var[X]=7.425.", "meta": {"id": "WUA_STAT_25K_008572", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.89, μ=6.26, σ=22.27, compute z."}]}, "response": "z = 1.375.", "meta": {"id": "WUA_STAT_25K_008573", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.14, μ=38.62, σ=22.16, compute z."}]}, "response": "z = 2.551.", "meta": {"id": "WUA_STAT_25K_008574", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.47, s=10.21, n=169, compute CI."}]}, "response": "95% CI ≈ [133.93, 137.01].", "meta": {"id": "WUA_STAT_25K_008575", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.101, sens=0.96, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.729.", "meta": {"id": "WUA_STAT_25K_008576", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.105, sens=0.94, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.344.", "meta": {"id": "WUA_STAT_25K_008577", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.86, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.664.", "meta": {"id": "WUA_STAT_25K_008578", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.49, s=17.22, n=54, compute CI."}]}, "response": "95% CI ≈ [137.90, 147.08].", "meta": {"id": "WUA_STAT_25K_008579", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.71, compute E and Var."}]}, "response": "E[X]=26.27; Var[X]=7.618.", "meta": {"id": "WUA_STAT_25K_008580", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.091, sens=0.87, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.635.", "meta": {"id": "WUA_STAT_25K_008581", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.77, compute E and Var."}]}, "response": "E[X]=4.62; Var[X]=1.063.", "meta": {"id": "WUA_STAT_25K_008582", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.24, s=27.18, n=165, compute CI."}]}, "response": "95% CI ≈ [11.09, 19.39].", "meta": {"id": "WUA_STAT_25K_008583", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.79, compute E and Var."}]}, "response": "E[X]=7.90; Var[X]=1.659.", "meta": {"id": "WUA_STAT_25K_008584", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=94.72, s=28.88, n=100, compute CI."}]}, "response": "95% CI ≈ [89.06, 100.38].", "meta": {"id": "WUA_STAT_25K_008585", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.34, μ=29.87, σ=17.39, compute z."}]}, "response": "z = 0.602.", "meta": {"id": "WUA_STAT_25K_008586", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.83, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.535.", "meta": {"id": "WUA_STAT_25K_008587", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.57, μ=71.82, σ=3.94, compute z."}]}, "response": "z = -3.617.", "meta": {"id": "WUA_STAT_25K_008588", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.3, μ=41.05, σ=12.02, compute z."}]}, "response": "z = 0.437.", "meta": {"id": "WUA_STAT_25K_008589", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.6, compute E and Var."}]}, "response": "E[X]=21.00; Var[X]=8.400.", "meta": {"id": "WUA_STAT_25K_008590", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.37, μ=70.3, σ=23.87, compute z."}]}, "response": "z = -1.882.", "meta": {"id": "WUA_STAT_25K_008591", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=143.39, s=19.97, n=34, compute CI."}]}, "response": "95% CI ≈ [136.68, 150.10].", "meta": {"id": "WUA_STAT_25K_008592", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.06, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.974.", "meta": {"id": "WUA_STAT_25K_008593", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.232, sens=0.98, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.514.", "meta": {"id": "WUA_STAT_25K_008594", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.75, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.441.", "meta": {"id": "WUA_STAT_25K_008595", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.72, compute E and Var."}]}, "response": "E[X]=18.00; Var[X]=5.040.", "meta": {"id": "WUA_STAT_25K_008596", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.73, μ=0.56, σ=17.76, compute z."}]}, "response": "z = 4.627.", "meta": {"id": "WUA_STAT_25K_008597", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.77, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.135.", "meta": {"id": "WUA_STAT_25K_008598", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.17, μ=22.23, σ=6.38, compute z."}]}, "response": "z = 4.379.", "meta": {"id": "WUA_STAT_25K_008599", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.37, compute E and Var."}]}, "response": "E[X]=14.43; Var[X]=9.091.", "meta": {"id": "WUA_STAT_25K_008600", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.01, s=2.86, n=151, compute CI."}]}, "response": "95% CI ≈ [143.55, 144.47].", "meta": {"id": "WUA_STAT_25K_008601", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.78, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.551.", "meta": {"id": "WUA_STAT_25K_008602", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.62, s=28.46, n=85, compute CI."}]}, "response": "95% CI ≈ [71.57, 83.67].", "meta": {"id": "WUA_STAT_25K_008603", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.72, μ=76.69, σ=21.35, compute z."}]}, "response": "z = -2.387.", "meta": {"id": "WUA_STAT_25K_008604", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.14, s=8.94, n=45, compute CI."}]}, "response": "95% CI ≈ [190.53, 195.75].", "meta": {"id": "WUA_STAT_25K_008605", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.29, compute E and Var."}]}, "response": "E[X]=7.25; Var[X]=5.147.", "meta": {"id": "WUA_STAT_25K_008606", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.31, μ=31.22, σ=8.33, compute z."}]}, "response": "z = -0.109.", "meta": {"id": "WUA_STAT_25K_008607", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.74, compute E and Var."}]}, "response": "E[X]=11.84; Var[X]=3.078.", "meta": {"id": "WUA_STAT_25K_008608", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.8, μ=72.89, σ=24.8, compute z."}]}, "response": "z = -2.746.", "meta": {"id": "WUA_STAT_25K_008609", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.57, s=7.47, n=163, compute CI."}]}, "response": "95% CI ≈ [187.42, 189.72].", "meta": {"id": "WUA_STAT_25K_008610", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.54, s=24.37, n=129, compute CI."}]}, "response": "95% CI ≈ [58.33, 66.75].", "meta": {"id": "WUA_STAT_25K_008611", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.09, compute E and Var."}]}, "response": "E[X]=3.06; Var[X]=2.785.", "meta": {"id": "WUA_STAT_25K_008612", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.7, μ=11.94, σ=14.6, compute z."}]}, "response": "z = 3.682.", "meta": {"id": "WUA_STAT_25K_008613", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.38, μ=26.94, σ=17.97, compute z."}]}, "response": "z = -0.198.", "meta": {"id": "WUA_STAT_25K_008614", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.47, μ=33.48, σ=5.79, compute z."}]}, "response": "z = 1.380.", "meta": {"id": "WUA_STAT_25K_008615", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.81, μ=62.03, σ=11.21, compute z."}]}, "response": "z = 1.854.", "meta": {"id": "WUA_STAT_25K_008616", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.65, μ=79.52, σ=22.45, compute z."}]}, "response": "z = -0.172.", "meta": {"id": "WUA_STAT_25K_008617", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.93, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.385.", "meta": {"id": "WUA_STAT_25K_008618", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.148, sens=0.76, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.868.", "meta": {"id": "WUA_STAT_25K_008619", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.78, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.636.", "meta": {"id": "WUA_STAT_25K_008620", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.11, s=33.82, n=75, compute CI."}]}, "response": "95% CI ≈ [17.46, 32.76].", "meta": {"id": "WUA_STAT_25K_008621", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.94, μ=68.57, σ=22.42, compute z."}]}, "response": "z = 1.176.", "meta": {"id": "WUA_STAT_25K_008622", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.096, sens=0.77, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.246.", "meta": {"id": "WUA_STAT_25K_008623", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.08, compute E and Var."}]}, "response": "E[X]=1.44; Var[X]=1.325.", "meta": {"id": "WUA_STAT_25K_008624", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.66, s=14.92, n=134, compute CI."}]}, "response": "95% CI ≈ [56.13, 61.19].", "meta": {"id": "WUA_STAT_25K_008625", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.44, s=8.67, n=101, compute CI."}]}, "response": "95% CI ≈ [51.75, 55.13].", "meta": {"id": "WUA_STAT_25K_008626", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.84, s=2.7, n=168, compute CI."}]}, "response": "95% CI ≈ [178.43, 179.25].", "meta": {"id": "WUA_STAT_25K_008627", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.67, compute E and Var."}]}, "response": "E[X]=22.78; Var[X]=7.517.", "meta": {"id": "WUA_STAT_25K_008628", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=108.82, s=6.61, n=80, compute CI."}]}, "response": "95% CI ≈ [107.37, 110.27].", "meta": {"id": "WUA_STAT_25K_008629", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.68, compute E and Var."}]}, "response": "E[X]=19.72; Var[X]=6.310.", "meta": {"id": "WUA_STAT_25K_008630", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.22, s=35.93, n=73, compute CI."}]}, "response": "95% CI ≈ [29.98, 46.46].", "meta": {"id": "WUA_STAT_25K_008631", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.09, compute E and Var."}]}, "response": "E[X]=1.44; Var[X]=1.310.", "meta": {"id": "WUA_STAT_25K_008632", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.22, compute E and Var."}]}, "response": "E[X]=2.42; Var[X]=1.888.", "meta": {"id": "WUA_STAT_25K_008633", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.92, μ=91.44, σ=4.58, compute z."}]}, "response": "z = -8.629.", "meta": {"id": "WUA_STAT_25K_008634", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.19, s=11.19, n=33, compute CI."}]}, "response": "95% CI ≈ [19.37, 27.01].", "meta": {"id": "WUA_STAT_25K_008635", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.9, s=34.28, n=79, compute CI."}]}, "response": "95% CI ≈ [70.34, 85.46].", "meta": {"id": "WUA_STAT_25K_008636", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.22, compute E and Var."}]}, "response": "E[X]=7.70; Var[X]=6.006.", "meta": {"id": "WUA_STAT_25K_008637", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.33, compute E and Var."}]}, "response": "E[X]=9.24; Var[X]=6.191.", "meta": {"id": "WUA_STAT_25K_008638", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.31, μ=8.9, σ=1.28, compute z."}]}, "response": "z = 23.758.", "meta": {"id": "WUA_STAT_25K_008639", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.053, sens=0.95, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.165.", "meta": {"id": "WUA_STAT_25K_008640", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=169.48, s=12.82, n=44, compute CI."}]}, "response": "95% CI ≈ [165.69, 173.27].", "meta": {"id": "WUA_STAT_25K_008641", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.78, s=6.54, n=170, compute CI."}]}, "response": "95% CI ≈ [58.80, 60.76].", "meta": {"id": "WUA_STAT_25K_008642", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.17, μ=23.88, σ=24.61, compute z."}]}, "response": "z = 1.150.", "meta": {"id": "WUA_STAT_25K_008643", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.104, sens=0.92, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.451.", "meta": {"id": "WUA_STAT_25K_008644", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.32, compute E and Var."}]}, "response": "E[X]=5.44; Var[X]=3.699.", "meta": {"id": "WUA_STAT_25K_008645", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.099, sens=0.8, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.355.", "meta": {"id": "WUA_STAT_25K_008646", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.54, μ=29.78, σ=24.36, compute z."}]}, "response": "z = -0.092.", "meta": {"id": "WUA_STAT_25K_008647", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.58, compute E and Var."}]}, "response": "E[X]=11.60; Var[X]=4.872.", "meta": {"id": "WUA_STAT_25K_008648", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.94, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.485.", "meta": {"id": "WUA_STAT_25K_008649", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.31, s=35.52, n=56, compute CI."}]}, "response": "95% CI ≈ [180.01, 198.61].", "meta": {"id": "WUA_STAT_25K_008650", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.93, μ=41.52, σ=2.37, compute z."}]}, "response": "z = 23.802.", "meta": {"id": "WUA_STAT_25K_008651", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.015, sens=0.71, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.265.", "meta": {"id": "WUA_STAT_25K_008652", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=47.82, s=1.93, n=55, compute CI."}]}, "response": "95% CI ≈ [47.31, 48.33].", "meta": {"id": "WUA_STAT_25K_008653", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.58, μ=46.6, σ=6.58, compute z."}]}, "response": "z = -0.459.", "meta": {"id": "WUA_STAT_25K_008654", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.058, sens=0.78, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.161.", "meta": {"id": "WUA_STAT_25K_008655", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.13, s=28.56, n=42, compute CI."}]}, "response": "95% CI ≈ [161.49, 178.77].", "meta": {"id": "WUA_STAT_25K_008656", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.01, μ=41.71, σ=7.27, compute z."}]}, "response": "z = 4.718.", "meta": {"id": "WUA_STAT_25K_008657", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.79, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.823.", "meta": {"id": "WUA_STAT_25K_008658", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.58, compute E and Var."}]}, "response": "E[X]=22.04; Var[X]=9.257.", "meta": {"id": "WUA_STAT_25K_008659", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.34, compute E and Var."}]}, "response": "E[X]=5.10; Var[X]=3.366.", "meta": {"id": "WUA_STAT_25K_008660", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.74, compute E and Var."}]}, "response": "E[X]=23.68; Var[X]=6.157.", "meta": {"id": "WUA_STAT_25K_008661", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.97, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.333.", "meta": {"id": "WUA_STAT_25K_008662", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.3, compute E and Var."}]}, "response": "E[X]=9.60; Var[X]=6.720.", "meta": {"id": "WUA_STAT_25K_008663", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.84, μ=73.04, σ=9.15, compute z."}]}, "response": "z = -2.863.", "meta": {"id": "WUA_STAT_25K_008664", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.73, s=15.15, n=16, compute CI."}]}, "response": "95% CI ≈ [129.31, 144.15].", "meta": {"id": "WUA_STAT_25K_008665", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.197, sens=0.71, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.466.", "meta": {"id": "WUA_STAT_25K_008666", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.17, compute E and Var."}]}, "response": "E[X]=3.40; Var[X]=2.822.", "meta": {"id": "WUA_STAT_25K_008667", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.22, μ=25.72, σ=11.0, compute z."}]}, "response": "z = -1.136.", "meta": {"id": "WUA_STAT_25K_008668", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.086, sens=0.88, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.235.", "meta": {"id": "WUA_STAT_25K_008669", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.06, μ=21.07, σ=18.01, compute z."}]}, "response": "z = 1.554.", "meta": {"id": "WUA_STAT_25K_008670", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.59, s=30.83, n=74, compute CI."}]}, "response": "95% CI ≈ [107.57, 121.61].", "meta": {"id": "WUA_STAT_25K_008671", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.48, μ=58.88, σ=7.31, compute z."}]}, "response": "z = -1.149.", "meta": {"id": "WUA_STAT_25K_008672", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.077, sens=0.8, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.260.", "meta": {"id": "WUA_STAT_25K_008673", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.77, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.328.", "meta": {"id": "WUA_STAT_25K_008674", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=68.3, s=5.3, n=41, compute CI."}]}, "response": "95% CI ≈ [66.68, 69.92].", "meta": {"id": "WUA_STAT_25K_008675", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.33, compute E and Var."}]}, "response": "E[X]=6.60; Var[X]=4.422.", "meta": {"id": "WUA_STAT_25K_008676", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=171.13, s=13.4, n=125, compute CI."}]}, "response": "95% CI ≈ [168.78, 173.48].", "meta": {"id": "WUA_STAT_25K_008677", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.53, s=33.12, n=35, compute CI."}]}, "response": "95% CI ≈ [116.56, 138.50].", "meta": {"id": "WUA_STAT_25K_008678", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.44, s=19.07, n=123, compute CI."}]}, "response": "95% CI ≈ [119.07, 125.81].", "meta": {"id": "WUA_STAT_25K_008679", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.04, s=15.14, n=60, compute CI."}]}, "response": "95% CI ≈ [13.21, 20.87].", "meta": {"id": "WUA_STAT_25K_008680", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.85, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.576.", "meta": {"id": "WUA_STAT_25K_008681", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.65, s=5.56, n=123, compute CI."}]}, "response": "95% CI ≈ [174.67, 176.63].", "meta": {"id": "WUA_STAT_25K_008682", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.61, compute E and Var."}]}, "response": "E[X]=10.37; Var[X]=4.044.", "meta": {"id": "WUA_STAT_25K_008683", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=171.23, s=4.69, n=38, compute CI."}]}, "response": "95% CI ≈ [169.74, 172.72].", "meta": {"id": "WUA_STAT_25K_008684", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.71, s=3.77, n=136, compute CI."}]}, "response": "95% CI ≈ [67.08, 68.34].", "meta": {"id": "WUA_STAT_25K_008685", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=94.56, s=29.28, n=135, compute CI."}]}, "response": "95% CI ≈ [89.62, 99.50].", "meta": {"id": "WUA_STAT_25K_008686", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.17, μ=26.33, σ=11.6, compute z."}]}, "response": "z = 0.503.", "meta": {"id": "WUA_STAT_25K_008687", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.43, compute E and Var."}]}, "response": "E[X]=15.91; Var[X]=9.069.", "meta": {"id": "WUA_STAT_25K_008688", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.57, compute E and Var."}]}, "response": "E[X]=14.82; Var[X]=6.373.", "meta": {"id": "WUA_STAT_25K_008689", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.209, sens=0.95, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.473.", "meta": {"id": "WUA_STAT_25K_008690", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.45, μ=16.49, σ=14.23, compute z."}]}, "response": "z = 1.754.", "meta": {"id": "WUA_STAT_25K_008691", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.69, s=7.09, n=160, compute CI."}]}, "response": "95% CI ≈ [84.59, 86.79].", "meta": {"id": "WUA_STAT_25K_008692", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.117, sens=0.91, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.388.", "meta": {"id": "WUA_STAT_25K_008693", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.29, compute E and Var."}]}, "response": "E[X]=4.06; Var[X]=2.883.", "meta": {"id": "WUA_STAT_25K_008694", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.58, μ=26.32, σ=18.25, compute z."}]}, "response": "z = 3.850.", "meta": {"id": "WUA_STAT_25K_008695", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.77, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.460.", "meta": {"id": "WUA_STAT_25K_008696", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.14, compute E and Var."}]}, "response": "E[X]=3.22; Var[X]=2.769.", "meta": {"id": "WUA_STAT_25K_008697", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.93, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.267.", "meta": {"id": "WUA_STAT_25K_008698", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.71, compute E and Var."}]}, "response": "E[X]=19.88; Var[X]=5.765.", "meta": {"id": "WUA_STAT_25K_008699", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.93, μ=11.47, σ=13.98, compute z."}]}, "response": "z = 0.605.", "meta": {"id": "WUA_STAT_25K_008700", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.7, μ=39.78, σ=17.07, compute z."}]}, "response": "z = -0.180.", "meta": {"id": "WUA_STAT_25K_008701", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.13, s=15.45, n=166, compute CI."}]}, "response": "95% CI ≈ [57.78, 62.48].", "meta": {"id": "WUA_STAT_25K_008702", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.4, s=34.44, n=21, compute CI."}]}, "response": "95% CI ≈ [54.67, 84.13].", "meta": {"id": "WUA_STAT_25K_008703", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.56, μ=36.51, σ=3.66, compute z."}]}, "response": "z = 9.577.", "meta": {"id": "WUA_STAT_25K_008704", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.13, compute E and Var."}]}, "response": "E[X]=2.73; Var[X]=2.375.", "meta": {"id": "WUA_STAT_25K_008705", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.74, μ=91.51, σ=7.14, compute z."}]}, "response": "z = -2.069.", "meta": {"id": "WUA_STAT_25K_008706", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.61, compute E and Var."}]}, "response": "E[X]=4.27; Var[X]=1.665.", "meta": {"id": "WUA_STAT_25K_008707", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.81, s=38.99, n=21, compute CI."}]}, "response": "95% CI ≈ [117.13, 150.49].", "meta": {"id": "WUA_STAT_25K_008708", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.09, μ=0.44, σ=2.69, compute z."}]}, "response": "z = 31.468.", "meta": {"id": "WUA_STAT_25K_008709", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.92, μ=32.32, σ=20.3, compute z."}]}, "response": "z = -0.562.", "meta": {"id": "WUA_STAT_25K_008710", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.35, μ=14.69, σ=4.87, compute z."}]}, "response": "z = -2.739.", "meta": {"id": "WUA_STAT_25K_008711", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.176, sens=0.95, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.670.", "meta": {"id": "WUA_STAT_25K_008712", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.78, compute E and Var."}]}, "response": "E[X]=14.04; Var[X]=3.089.", "meta": {"id": "WUA_STAT_25K_008713", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.38, compute E and Var."}]}, "response": "E[X]=9.88; Var[X]=6.126.", "meta": {"id": "WUA_STAT_25K_008714", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=141.62, s=39.83, n=101, compute CI."}]}, "response": "95% CI ≈ [133.85, 149.39].", "meta": {"id": "WUA_STAT_25K_008715", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.71, compute E and Var."}]}, "response": "E[X]=26.98; Var[X]=7.824.", "meta": {"id": "WUA_STAT_25K_008716", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.58, compute E and Var."}]}, "response": "E[X]=12.76; Var[X]=5.359.", "meta": {"id": "WUA_STAT_25K_008717", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.176, sens=0.92, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.404.", "meta": {"id": "WUA_STAT_25K_008718", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.16, s=17.45, n=84, compute CI."}]}, "response": "95% CI ≈ [85.43, 92.89].", "meta": {"id": "WUA_STAT_25K_008719", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.65, μ=28.99, σ=5.91, compute z."}]}, "response": "z = 5.695.", "meta": {"id": "WUA_STAT_25K_008720", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.36, compute E and Var."}]}, "response": "E[X]=8.64; Var[X]=5.530.", "meta": {"id": "WUA_STAT_25K_008721", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.34, s=26.79, n=102, compute CI."}]}, "response": "95% CI ≈ [85.14, 95.54].", "meta": {"id": "WUA_STAT_25K_008722", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=168.89, s=26.4, n=196, compute CI."}]}, "response": "95% CI ≈ [165.19, 172.59].", "meta": {"id": "WUA_STAT_25K_008723", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.026, sens=0.71, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.079.", "meta": {"id": "WUA_STAT_25K_008724", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.21, compute E and Var."}]}, "response": "E[X]=7.35; Var[X]=5.806.", "meta": {"id": "WUA_STAT_25K_008725", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.115, sens=0.89, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.452.", "meta": {"id": "WUA_STAT_25K_008726", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.66, s=15.34, n=164, compute CI."}]}, "response": "95% CI ≈ [38.31, 43.01].", "meta": {"id": "WUA_STAT_25K_008727", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.72, μ=47.63, σ=19.02, compute z."}]}, "response": "z = 0.741.", "meta": {"id": "WUA_STAT_25K_008728", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=64.61, s=36.71, n=194, compute CI."}]}, "response": "95% CI ≈ [59.44, 69.78].", "meta": {"id": "WUA_STAT_25K_008729", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=71.11, s=30.32, n=104, compute CI."}]}, "response": "95% CI ≈ [65.28, 76.94].", "meta": {"id": "WUA_STAT_25K_008730", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.95, μ=37.99, σ=17.37, compute z."}]}, "response": "z = -1.557.", "meta": {"id": "WUA_STAT_25K_008731", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.2, compute E and Var."}]}, "response": "E[X]=3.00; Var[X]=2.400.", "meta": {"id": "WUA_STAT_25K_008732", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.63, μ=9.1, σ=18.2, compute z."}]}, "response": "z = -0.191.", "meta": {"id": "WUA_STAT_25K_008733", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.27, compute E and Var."}]}, "response": "E[X]=4.86; Var[X]=3.548.", "meta": {"id": "WUA_STAT_25K_008734", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.55, compute E and Var."}]}, "response": "E[X]=14.30; Var[X]=6.435.", "meta": {"id": "WUA_STAT_25K_008735", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.38, compute E and Var."}]}, "response": "E[X]=4.18; Var[X]=2.592.", "meta": {"id": "WUA_STAT_25K_008736", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.12, μ=62.93, σ=7.15, compute z."}]}, "response": "z = -0.113.", "meta": {"id": "WUA_STAT_25K_008737", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.241, sens=0.76, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.650.", "meta": {"id": "WUA_STAT_25K_008738", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.57, s=30.76, n=137, compute CI."}]}, "response": "95% CI ≈ [79.42, 89.72].", "meta": {"id": "WUA_STAT_25K_008739", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.73, μ=16.41, σ=9.71, compute z."}]}, "response": "z = 7.448.", "meta": {"id": "WUA_STAT_25K_008740", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.7, s=17.02, n=75, compute CI."}]}, "response": "95% CI ≈ [76.85, 84.55].", "meta": {"id": "WUA_STAT_25K_008741", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.05, compute E and Var."}]}, "response": "E[X]=0.60; Var[X]=0.570.", "meta": {"id": "WUA_STAT_25K_008742", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.78, μ=89.46, σ=14.78, compute z."}]}, "response": "z = -4.241.", "meta": {"id": "WUA_STAT_25K_008743", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.73, μ=5.81, σ=14.19, compute z."}]}, "response": "z = 4.293.", "meta": {"id": "WUA_STAT_25K_008744", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.27, μ=87.22, σ=18.55, compute z."}]}, "response": "z = -1.237.", "meta": {"id": "WUA_STAT_25K_008745", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.089, sens=0.89, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.635.", "meta": {"id": "WUA_STAT_25K_008746", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.58, compute E and Var."}]}, "response": "E[X]=5.22; Var[X]=2.192.", "meta": {"id": "WUA_STAT_25K_008747", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.49, μ=53.95, σ=19.2, compute z."}]}, "response": "z = 1.226.", "meta": {"id": "WUA_STAT_25K_008748", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.233, sens=0.8, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.536.", "meta": {"id": "WUA_STAT_25K_008749", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.7, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.578.", "meta": {"id": "WUA_STAT_25K_008750", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.26, s=13.17, n=119, compute CI."}]}, "response": "95% CI ≈ [18.89, 23.63].", "meta": {"id": "WUA_STAT_25K_008751", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.77, μ=40.71, σ=8.82, compute z."}]}, "response": "z = 4.995.", "meta": {"id": "WUA_STAT_25K_008752", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.6, compute E and Var."}]}, "response": "E[X]=22.20; Var[X]=8.880.", "meta": {"id": "WUA_STAT_25K_008753", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.69, s=22.74, n=107, compute CI."}]}, "response": "95% CI ≈ [185.38, 194.00].", "meta": {"id": "WUA_STAT_25K_008754", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.8, s=7.81, n=111, compute CI."}]}, "response": "95% CI ≈ [122.35, 125.25].", "meta": {"id": "WUA_STAT_25K_008755", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.27, s=9.84, n=190, compute CI."}]}, "response": "95% CI ≈ [44.87, 47.67].", "meta": {"id": "WUA_STAT_25K_008756", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=104.18, s=17.0, n=84, compute CI."}]}, "response": "95% CI ≈ [100.54, 107.82].", "meta": {"id": "WUA_STAT_25K_008757", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.62, compute E and Var."}]}, "response": "E[X]=21.08; Var[X]=8.010.", "meta": {"id": "WUA_STAT_25K_008758", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.53, μ=22.19, σ=14.53, compute z."}]}, "response": "z = 2.501.", "meta": {"id": "WUA_STAT_25K_008759", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.36, s=19.21, n=39, compute CI."}]}, "response": "95% CI ≈ [172.33, 184.39].", "meta": {"id": "WUA_STAT_25K_008760", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.77, compute E and Var."}]}, "response": "E[X]=26.18; Var[X]=6.021.", "meta": {"id": "WUA_STAT_25K_008761", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.62, compute E and Var."}]}, "response": "E[X]=6.20; Var[X]=2.356.", "meta": {"id": "WUA_STAT_25K_008762", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.78, compute E and Var."}]}, "response": "E[X]=20.28; Var[X]=4.462.", "meta": {"id": "WUA_STAT_25K_008763", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.45, compute E and Var."}]}, "response": "E[X]=7.65; Var[X]=4.208.", "meta": {"id": "WUA_STAT_25K_008764", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.32, s=19.84, n=149, compute CI."}]}, "response": "95% CI ≈ [58.13, 64.51].", "meta": {"id": "WUA_STAT_25K_008765", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.76, s=10.59, n=114, compute CI."}]}, "response": "95% CI ≈ [158.82, 162.70].", "meta": {"id": "WUA_STAT_25K_008766", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.055, sens=0.73, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.124.", "meta": {"id": "WUA_STAT_25K_008767", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.68, s=37.34, n=98, compute CI."}]}, "response": "95% CI ≈ [79.29, 94.07].", "meta": {"id": "WUA_STAT_25K_008768", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.91, s=35.81, n=31, compute CI."}]}, "response": "95% CI ≈ [86.30, 111.52].", "meta": {"id": "WUA_STAT_25K_008769", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.9, μ=55.38, σ=4.55, compute z."}]}, "response": "z = 0.334.", "meta": {"id": "WUA_STAT_25K_008770", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.12, compute E and Var."}]}, "response": "E[X]=2.64; Var[X]=2.323.", "meta": {"id": "WUA_STAT_25K_008771", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.11, s=16.73, n=140, compute CI."}]}, "response": "95% CI ≈ [171.34, 176.88].", "meta": {"id": "WUA_STAT_25K_008772", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.165, sens=0.78, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.885.", "meta": {"id": "WUA_STAT_25K_008773", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.168, sens=0.88, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.717.", "meta": {"id": "WUA_STAT_25K_008774", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.79, μ=0.51, σ=1.75, compute z."}]}, "response": "z = 28.160.", "meta": {"id": "WUA_STAT_25K_008775", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.14, μ=63.75, σ=5.97, compute z."}]}, "response": "z = -8.142.", "meta": {"id": "WUA_STAT_25K_008776", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.61, compute E and Var."}]}, "response": "E[X]=18.91; Var[X]=7.375.", "meta": {"id": "WUA_STAT_25K_008777", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.44, compute E and Var."}]}, "response": "E[X]=7.04; Var[X]=3.942.", "meta": {"id": "WUA_STAT_25K_008778", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.94, s=13.57, n=132, compute CI."}]}, "response": "95% CI ≈ [58.63, 63.25].", "meta": {"id": "WUA_STAT_25K_008779", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.124, sens=0.95, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.871.", "meta": {"id": "WUA_STAT_25K_008780", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.79, s=26.69, n=24, compute CI."}]}, "response": "95% CI ≈ [28.11, 49.47].", "meta": {"id": "WUA_STAT_25K_008781", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.68, μ=48.75, σ=2.99, compute z."}]}, "response": "z = -5.709.", "meta": {"id": "WUA_STAT_25K_008782", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.3, μ=36.67, σ=4.75, compute z."}]}, "response": "z = -3.867.", "meta": {"id": "WUA_STAT_25K_008783", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.34, compute E and Var."}]}, "response": "E[X]=4.76; Var[X]=3.142.", "meta": {"id": "WUA_STAT_25K_008784", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.2, μ=69.83, σ=6.81, compute z."}]}, "response": "z = 2.844.", "meta": {"id": "WUA_STAT_25K_008785", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.69, compute E and Var."}]}, "response": "E[X]=10.35; Var[X]=3.209.", "meta": {"id": "WUA_STAT_25K_008786", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.7, s=26.25, n=177, compute CI."}]}, "response": "95% CI ≈ [186.83, 194.57].", "meta": {"id": "WUA_STAT_25K_008787", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.44, μ=64.87, σ=12.83, compute z."}]}, "response": "z = -3.931.", "meta": {"id": "WUA_STAT_25K_008788", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.83, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.382.", "meta": {"id": "WUA_STAT_25K_008789", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.5, compute E and Var."}]}, "response": "E[X]=16.50; Var[X]=8.250.", "meta": {"id": "WUA_STAT_25K_008790", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.89, μ=49.24, σ=21.41, compute z."}]}, "response": "z = -0.156.", "meta": {"id": "WUA_STAT_25K_008791", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.2, compute E and Var."}]}, "response": "E[X]=3.00; Var[X]=2.400.", "meta": {"id": "WUA_STAT_25K_008792", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.51, compute E and Var."}]}, "response": "E[X]=3.57; Var[X]=1.749.", "meta": {"id": "WUA_STAT_25K_008793", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.031, sens=0.92, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.113.", "meta": {"id": "WUA_STAT_25K_008794", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.03, μ=43.42, σ=7.02, compute z."}]}, "response": "z = -3.047.", "meta": {"id": "WUA_STAT_25K_008795", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.057, sens=0.71, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.137.", "meta": {"id": "WUA_STAT_25K_008796", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.217, sens=0.81, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.818.", "meta": {"id": "WUA_STAT_25K_008797", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.29, compute E and Var."}]}, "response": "E[X]=4.64; Var[X]=3.294.", "meta": {"id": "WUA_STAT_25K_008798", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.99, μ=26.65, σ=7.37, compute z."}]}, "response": "z = 2.760.", "meta": {"id": "WUA_STAT_25K_008799", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.22, μ=81.57, σ=14.64, compute z."}]}, "response": "z = 1.001.", "meta": {"id": "WUA_STAT_25K_008800", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.12, compute E and Var."}]}, "response": "E[X]=0.96; Var[X]=0.845.", "meta": {"id": "WUA_STAT_25K_008801", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.18, s=33.7, n=138, compute CI."}]}, "response": "95% CI ≈ [29.56, 40.80].", "meta": {"id": "WUA_STAT_25K_008802", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.31, s=1.54, n=34, compute CI."}]}, "response": "95% CI ≈ [138.79, 139.83].", "meta": {"id": "WUA_STAT_25K_008803", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.062, sens=0.71, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.540.", "meta": {"id": "WUA_STAT_25K_008804", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.91, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.470.", "meta": {"id": "WUA_STAT_25K_008805", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.69, μ=4.2, σ=10.23, compute z."}]}, "response": "z = 3.762.", "meta": {"id": "WUA_STAT_25K_008806", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.07, sens=0.91, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.345.", "meta": {"id": "WUA_STAT_25K_008807", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.68, compute E and Var."}]}, "response": "E[X]=26.52; Var[X]=8.486.", "meta": {"id": "WUA_STAT_25K_008808", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.06, s=12.75, n=72, compute CI."}]}, "response": "95% CI ≈ [31.11, 37.01].", "meta": {"id": "WUA_STAT_25K_008809", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.8, s=11.79, n=199, compute CI."}]}, "response": "95% CI ≈ [61.16, 64.44].", "meta": {"id": "WUA_STAT_25K_008810", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.12, compute E and Var."}]}, "response": "E[X]=0.60; Var[X]=0.528.", "meta": {"id": "WUA_STAT_25K_008811", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.83, μ=37.77, σ=22.32, compute z."}]}, "response": "z = 1.033.", "meta": {"id": "WUA_STAT_25K_008812", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=28.38, s=27.68, n=117, compute CI."}]}, "response": "95% CI ≈ [23.36, 33.40].", "meta": {"id": "WUA_STAT_25K_008813", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.134, sens=0.79, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.859.", "meta": {"id": "WUA_STAT_25K_008814", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.162, sens=0.74, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.338.", "meta": {"id": "WUA_STAT_25K_008815", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.73, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.409.", "meta": {"id": "WUA_STAT_25K_008816", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.214, sens=0.95, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.603.", "meta": {"id": "WUA_STAT_25K_008817", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.57, μ=35.16, σ=21.13, compute z."}]}, "response": "z = -1.637.", "meta": {"id": "WUA_STAT_25K_008818", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.55, compute E and Var."}]}, "response": "E[X]=14.30; Var[X]=6.435.", "meta": {"id": "WUA_STAT_25K_008819", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.8, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.086.", "meta": {"id": "WUA_STAT_25K_008820", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.15, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.785.", "meta": {"id": "WUA_STAT_25K_008821", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.77, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.653.", "meta": {"id": "WUA_STAT_25K_008822", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.053, sens=0.89, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.499.", "meta": {"id": "WUA_STAT_25K_008823", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.64, compute E and Var."}]}, "response": "E[X]=12.80; Var[X]=4.608.", "meta": {"id": "WUA_STAT_25K_008824", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.08, μ=78.79, σ=24.89, compute z."}]}, "response": "z = -0.912.", "meta": {"id": "WUA_STAT_25K_008825", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.47, s=33.18, n=24, compute CI."}]}, "response": "95% CI ≈ [56.20, 82.74].", "meta": {"id": "WUA_STAT_25K_008826", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.37, μ=92.89, σ=19.39, compute z."}]}, "response": "z = -3.637.", "meta": {"id": "WUA_STAT_25K_008827", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.27, compute E and Var."}]}, "response": "E[X]=10.80; Var[X]=7.884.", "meta": {"id": "WUA_STAT_25K_008828", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.5, compute E and Var."}]}, "response": "E[X]=7.50; Var[X]=3.750.", "meta": {"id": "WUA_STAT_25K_008829", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.34, μ=21.69, σ=16.37, compute z."}]}, "response": "z = 0.467.", "meta": {"id": "WUA_STAT_25K_008830", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.4, μ=83.21, σ=18.1, compute z."}]}, "response": "z = -0.984.", "meta": {"id": "WUA_STAT_25K_008831", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.34, compute E and Var."}]}, "response": "E[X]=12.58; Var[X]=8.303.", "meta": {"id": "WUA_STAT_25K_008832", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.86, s=37.46, n=25, compute CI."}]}, "response": "95% CI ≈ [111.18, 140.54].", "meta": {"id": "WUA_STAT_25K_008833", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.3, compute E and Var."}]}, "response": "E[X]=6.90; Var[X]=4.830.", "meta": {"id": "WUA_STAT_25K_008834", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.37, μ=44.37, σ=1.79, compute z."}]}, "response": "z = -12.849.", "meta": {"id": "WUA_STAT_25K_008835", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.8, s=5.82, n=188, compute CI."}]}, "response": "95% CI ≈ [171.97, 173.63].", "meta": {"id": "WUA_STAT_25K_008836", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.209, sens=0.95, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.863.", "meta": {"id": "WUA_STAT_25K_008837", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.14, sens=0.96, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.635.", "meta": {"id": "WUA_STAT_25K_008838", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.67, compute E and Var."}]}, "response": "E[X]=22.78; Var[X]=7.517.", "meta": {"id": "WUA_STAT_25K_008839", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.72, s=19.89, n=180, compute CI."}]}, "response": "95% CI ≈ [28.81, 34.63].", "meta": {"id": "WUA_STAT_25K_008840", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.11, compute E and Var."}]}, "response": "E[X]=1.21; Var[X]=1.077.", "meta": {"id": "WUA_STAT_25K_008841", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.85, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.475.", "meta": {"id": "WUA_STAT_25K_008842", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.23, compute E and Var."}]}, "response": "E[X]=2.07; Var[X]=1.594.", "meta": {"id": "WUA_STAT_25K_008843", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.65, compute E and Var."}]}, "response": "E[X]=11.05; Var[X]=3.868.", "meta": {"id": "WUA_STAT_25K_008844", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.7, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=2.520.", "meta": {"id": "WUA_STAT_25K_008845", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.13, compute E and Var."}]}, "response": "E[X]=3.64; Var[X]=3.167.", "meta": {"id": "WUA_STAT_25K_008846", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.24, μ=72.91, σ=22.42, compute z."}]}, "response": "z = -0.922.", "meta": {"id": "WUA_STAT_25K_008847", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.37, μ=81.61, σ=24.8, compute z."}]}, "response": "z = 0.031.", "meta": {"id": "WUA_STAT_25K_008848", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.055, sens=0.97, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.261.", "meta": {"id": "WUA_STAT_25K_008849", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.79, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.137.", "meta": {"id": "WUA_STAT_25K_008850", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.66, μ=59.86, σ=4.73, compute z."}]}, "response": "z = -12.093.", "meta": {"id": "WUA_STAT_25K_008851", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.56, μ=46.84, σ=19.89, compute z."}]}, "response": "z = -1.774.", "meta": {"id": "WUA_STAT_25K_008852", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.49, s=19.07, n=55, compute CI."}]}, "response": "95% CI ≈ [165.45, 175.53].", "meta": {"id": "WUA_STAT_25K_008853", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.36, s=37.05, n=103, compute CI."}]}, "response": "95% CI ≈ [55.20, 69.52].", "meta": {"id": "WUA_STAT_25K_008854", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.41, compute E and Var."}]}, "response": "E[X]=4.51; Var[X]=2.661.", "meta": {"id": "WUA_STAT_25K_008855", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.47, s=24.42, n=56, compute CI."}]}, "response": "95% CI ≈ [172.07, 184.87].", "meta": {"id": "WUA_STAT_25K_008856", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.09, μ=18.57, σ=20.65, compute z."}]}, "response": "z = -0.895.", "meta": {"id": "WUA_STAT_25K_008857", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.92, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.720.", "meta": {"id": "WUA_STAT_25K_008858", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.16, compute E and Var."}]}, "response": "E[X]=0.96; Var[X]=0.806.", "meta": {"id": "WUA_STAT_25K_008859", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.44, compute E and Var."}]}, "response": "E[X]=14.52; Var[X]=8.131.", "meta": {"id": "WUA_STAT_25K_008860", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.15, compute E and Var."}]}, "response": "E[X]=1.95; Var[X]=1.657.", "meta": {"id": "WUA_STAT_25K_008861", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.31, compute E and Var."}]}, "response": "E[X]=4.65; Var[X]=3.208.", "meta": {"id": "WUA_STAT_25K_008862", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.87, s=28.56, n=54, compute CI."}]}, "response": "95% CI ≈ [154.25, 169.49].", "meta": {"id": "WUA_STAT_25K_008863", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.21, compute E and Var."}]}, "response": "E[X]=6.30; Var[X]=4.977.", "meta": {"id": "WUA_STAT_25K_008864", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.044, sens=0.81, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.293.", "meta": {"id": "WUA_STAT_25K_008865", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.26, s=8.63, n=180, compute CI."}]}, "response": "95% CI ≈ [132.00, 134.52].", "meta": {"id": "WUA_STAT_25K_008866", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.7, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.493.", "meta": {"id": "WUA_STAT_25K_008867", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.47, compute E and Var."}]}, "response": "E[X]=16.92; Var[X]=8.968.", "meta": {"id": "WUA_STAT_25K_008868", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.063, sens=0.86, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.224.", "meta": {"id": "WUA_STAT_25K_008869", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.79, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.330.", "meta": {"id": "WUA_STAT_25K_008870", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.088, sens=0.74, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.222.", "meta": {"id": "WUA_STAT_25K_008871", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.28, μ=39.51, σ=17.36, compute z."}]}, "response": "z = 1.484.", "meta": {"id": "WUA_STAT_25K_008872", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.111, sens=0.86, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.328.", "meta": {"id": "WUA_STAT_25K_008873", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.19, μ=79.27, σ=11.51, compute z."}]}, "response": "z = -3.743.", "meta": {"id": "WUA_STAT_25K_008874", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.131, sens=0.76, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.306.", "meta": {"id": "WUA_STAT_25K_008875", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.72, s=19.2, n=100, compute CI."}]}, "response": "95% CI ≈ [142.96, 150.48].", "meta": {"id": "WUA_STAT_25K_008876", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.54, compute E and Var."}]}, "response": "E[X]=15.12; Var[X]=6.955.", "meta": {"id": "WUA_STAT_25K_008877", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.44, compute E and Var."}]}, "response": "E[X]=14.52; Var[X]=8.131.", "meta": {"id": "WUA_STAT_25K_008878", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.01, s=21.31, n=111, compute CI."}]}, "response": "95% CI ≈ [13.05, 20.97].", "meta": {"id": "WUA_STAT_25K_008879", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=52.13, s=39.21, n=105, compute CI."}]}, "response": "95% CI ≈ [44.63, 59.63].", "meta": {"id": "WUA_STAT_25K_008880", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.103, sens=0.77, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.747.", "meta": {"id": "WUA_STAT_25K_008881", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.54, μ=81.87, σ=18.99, compute z."}]}, "response": "z = -2.808.", "meta": {"id": "WUA_STAT_25K_008882", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.45, μ=35.23, σ=18.42, compute z."}]}, "response": "z = -0.422.", "meta": {"id": "WUA_STAT_25K_008883", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.41, μ=30.16, σ=6.42, compute z."}]}, "response": "z = 8.762.", "meta": {"id": "WUA_STAT_25K_008884", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=72.74, s=39.11, n=33, compute CI."}]}, "response": "95% CI ≈ [59.40, 86.08].", "meta": {"id": "WUA_STAT_25K_008885", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.84, μ=29.22, σ=24.79, compute z."}]}, "response": "z = 0.509.", "meta": {"id": "WUA_STAT_25K_008886", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.51, μ=85.3, σ=12.11, compute z."}]}, "response": "z = -2.955.", "meta": {"id": "WUA_STAT_25K_008887", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.31, compute E and Var."}]}, "response": "E[X]=9.61; Var[X]=6.631.", "meta": {"id": "WUA_STAT_25K_008888", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.58, μ=50.53, σ=15.93, compute z."}]}, "response": "z = -2.257.", "meta": {"id": "WUA_STAT_25K_008889", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=108.78, s=11.59, n=135, compute CI."}]}, "response": "95% CI ≈ [106.82, 110.74].", "meta": {"id": "WUA_STAT_25K_008890", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.69, μ=81.68, σ=6.74, compute z."}]}, "response": "z = -4.895.", "meta": {"id": "WUA_STAT_25K_008891", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.33, s=16.55, n=156, compute CI."}]}, "response": "95% CI ≈ [73.73, 78.93].", "meta": {"id": "WUA_STAT_25K_008892", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=16.68, s=11.87, n=109, compute CI."}]}, "response": "95% CI ≈ [14.45, 18.91].", "meta": {"id": "WUA_STAT_25K_008893", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.59, μ=17.96, σ=4.85, compute z."}]}, "response": "z = 0.748.", "meta": {"id": "WUA_STAT_25K_008894", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.99, s=14.54, n=25, compute CI."}]}, "response": "95% CI ≈ [22.29, 33.69].", "meta": {"id": "WUA_STAT_25K_008895", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.64, μ=53.21, σ=20.3, compute z."}]}, "response": "z = 0.908.", "meta": {"id": "WUA_STAT_25K_008896", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.4, s=27.23, n=179, compute CI."}]}, "response": "95% CI ≈ [186.41, 194.39].", "meta": {"id": "WUA_STAT_25K_008897", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.72, compute E and Var."}]}, "response": "E[X]=7.92; Var[X]=2.218.", "meta": {"id": "WUA_STAT_25K_008898", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.09, s=12.67, n=152, compute CI."}]}, "response": "95% CI ≈ [170.08, 174.10].", "meta": {"id": "WUA_STAT_25K_008899", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.61, μ=39.47, σ=14.7, compute z."}]}, "response": "z = -2.167.", "meta": {"id": "WUA_STAT_25K_008900", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.1, s=29.94, n=49, compute CI."}]}, "response": "95% CI ≈ [80.72, 97.48].", "meta": {"id": "WUA_STAT_25K_008901", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.224, sens=0.75, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.625.", "meta": {"id": "WUA_STAT_25K_008902", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.99, s=31.08, n=121, compute CI."}]}, "response": "95% CI ≈ [124.45, 135.53].", "meta": {"id": "WUA_STAT_25K_008903", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.24, sens=0.84, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.930.", "meta": {"id": "WUA_STAT_25K_008904", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.29, s=5.61, n=33, compute CI."}]}, "response": "95% CI ≈ [34.38, 38.20].", "meta": {"id": "WUA_STAT_25K_008905", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.6, compute E and Var."}]}, "response": "E[X]=11.40; Var[X]=4.560.", "meta": {"id": "WUA_STAT_25K_008906", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.85, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.411.", "meta": {"id": "WUA_STAT_25K_008907", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.31, compute E and Var."}]}, "response": "E[X]=6.51; Var[X]=4.492.", "meta": {"id": "WUA_STAT_25K_008908", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.64, μ=26.13, σ=4.11, compute z."}]}, "response": "z = -4.012.", "meta": {"id": "WUA_STAT_25K_008909", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=155.68, s=17.58, n=113, compute CI."}]}, "response": "95% CI ≈ [152.44, 158.92].", "meta": {"id": "WUA_STAT_25K_008910", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.7, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.429.", "meta": {"id": "WUA_STAT_25K_008911", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.4, compute E and Var."}]}, "response": "E[X]=13.60; Var[X]=8.160.", "meta": {"id": "WUA_STAT_25K_008912", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.38, s=25.65, n=106, compute CI."}]}, "response": "95% CI ≈ [19.50, 29.26].", "meta": {"id": "WUA_STAT_25K_008913", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.6, compute E and Var."}]}, "response": "E[X]=3.60; Var[X]=1.440.", "meta": {"id": "WUA_STAT_25K_008914", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.66, compute E and Var."}]}, "response": "E[X]=14.52; Var[X]=4.937.", "meta": {"id": "WUA_STAT_25K_008915", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.182, sens=0.95, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.585.", "meta": {"id": "WUA_STAT_25K_008916", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.65, μ=79.89, σ=19.74, compute z."}]}, "response": "z = -3.508.", "meta": {"id": "WUA_STAT_25K_008917", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=39.17, s=16.43, n=62, compute CI."}]}, "response": "95% CI ≈ [35.08, 43.26].", "meta": {"id": "WUA_STAT_25K_008918", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.59, μ=62.04, σ=20.57, compute z."}]}, "response": "z = -0.022.", "meta": {"id": "WUA_STAT_25K_008919", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.31, s=14.49, n=38, compute CI."}]}, "response": "95% CI ≈ [176.70, 185.92].", "meta": {"id": "WUA_STAT_25K_008920", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.1, s=4.17, n=40, compute CI."}]}, "response": "95% CI ≈ [48.81, 51.39].", "meta": {"id": "WUA_STAT_25K_008921", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.121, sens=0.88, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.389.", "meta": {"id": "WUA_STAT_25K_008922", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.7, compute E and Var."}]}, "response": "E[X]=28.00; Var[X]=8.400.", "meta": {"id": "WUA_STAT_25K_008923", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.46, s=8.99, n=42, compute CI."}]}, "response": "95% CI ≈ [77.74, 83.18].", "meta": {"id": "WUA_STAT_25K_008924", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.08, μ=14.99, σ=20.7, compute z."}]}, "response": "z = 0.391.", "meta": {"id": "WUA_STAT_25K_008925", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.44, compute E and Var."}]}, "response": "E[X]=14.96; Var[X]=8.378.", "meta": {"id": "WUA_STAT_25K_008926", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.52, s=32.14, n=106, compute CI."}]}, "response": "95% CI ≈ [104.40, 116.64].", "meta": {"id": "WUA_STAT_25K_008927", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.223, sens=0.92, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.545.", "meta": {"id": "WUA_STAT_25K_008928", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.85, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.844.", "meta": {"id": "WUA_STAT_25K_008929", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.11, compute E and Var."}]}, "response": "E[X]=2.09; Var[X]=1.860.", "meta": {"id": "WUA_STAT_25K_008930", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.51, s=39.42, n=117, compute CI."}]}, "response": "95% CI ≈ [130.37, 144.65].", "meta": {"id": "WUA_STAT_25K_008931", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.179, sens=0.82, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.460.", "meta": {"id": "WUA_STAT_25K_008932", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.86, μ=93.95, σ=3.53, compute z."}]}, "response": "z = -13.340.", "meta": {"id": "WUA_STAT_25K_008933", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.07, s=38.15, n=59, compute CI."}]}, "response": "95% CI ≈ [65.34, 84.80].", "meta": {"id": "WUA_STAT_25K_008934", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.143, sens=0.79, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.320.", "meta": {"id": "WUA_STAT_25K_008935", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.45, compute E and Var."}]}, "response": "E[X]=14.85; Var[X]=8.168.", "meta": {"id": "WUA_STAT_25K_008936", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.94, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.460.", "meta": {"id": "WUA_STAT_25K_008937", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=26.59, s=4.41, n=162, compute CI."}]}, "response": "95% CI ≈ [25.91, 27.27].", "meta": {"id": "WUA_STAT_25K_008938", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.95, s=29.19, n=50, compute CI."}]}, "response": "95% CI ≈ [9.86, 26.04].", "meta": {"id": "WUA_STAT_25K_008939", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.15, μ=29.98, σ=11.68, compute z."}]}, "response": "z = 1.812.", "meta": {"id": "WUA_STAT_25K_008940", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=81.3, s=21.99, n=88, compute CI."}]}, "response": "95% CI ≈ [76.71, 85.89].", "meta": {"id": "WUA_STAT_25K_008941", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=197.61, s=20.68, n=156, compute CI."}]}, "response": "95% CI ≈ [194.36, 200.86].", "meta": {"id": "WUA_STAT_25K_008942", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.47, μ=70.84, σ=11.88, compute z."}]}, "response": "z = -3.146.", "meta": {"id": "WUA_STAT_25K_008943", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.94, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.565.", "meta": {"id": "WUA_STAT_25K_008944", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.91, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.091.", "meta": {"id": "WUA_STAT_25K_008945", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.48, compute E and Var."}]}, "response": "E[X]=13.92; Var[X]=7.238.", "meta": {"id": "WUA_STAT_25K_008946", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.24, μ=42.62, σ=14.0, compute z."}]}, "response": "z = 3.973.", "meta": {"id": "WUA_STAT_25K_008947", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.19, μ=20.8, σ=19.74, compute z."}]}, "response": "z = 2.198.", "meta": {"id": "WUA_STAT_25K_008948", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=191.73, s=10.41, n=96, compute CI."}]}, "response": "95% CI ≈ [189.65, 193.81].", "meta": {"id": "WUA_STAT_25K_008949", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.53, s=29.3, n=191, compute CI."}]}, "response": "95% CI ≈ [134.37, 142.69].", "meta": {"id": "WUA_STAT_25K_008950", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.75, s=28.78, n=130, compute CI."}]}, "response": "95% CI ≈ [36.80, 46.70].", "meta": {"id": "WUA_STAT_25K_008951", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.247, sens=0.76, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.926.", "meta": {"id": "WUA_STAT_25K_008952", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.38, s=9.56, n=21, compute CI."}]}, "response": "95% CI ≈ [11.29, 19.47].", "meta": {"id": "WUA_STAT_25K_008953", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.5, compute E and Var."}]}, "response": "E[X]=13.50; Var[X]=6.750.", "meta": {"id": "WUA_STAT_25K_008954", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.4, μ=19.28, σ=24.47, compute z."}]}, "response": "z = 0.863.", "meta": {"id": "WUA_STAT_25K_008955", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.7, compute E and Var."}]}, "response": "E[X]=18.90; Var[X]=5.670.", "meta": {"id": "WUA_STAT_25K_008956", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.82, s=9.58, n=99, compute CI."}]}, "response": "95% CI ≈ [192.93, 196.71].", "meta": {"id": "WUA_STAT_25K_008957", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.16, μ=96.53, σ=19.6, compute z."}]}, "response": "z = -2.621.", "meta": {"id": "WUA_STAT_25K_008958", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.89, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.498.", "meta": {"id": "WUA_STAT_25K_008959", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.56, s=35.42, n=13, compute CI."}]}, "response": "95% CI ≈ [57.31, 95.81].", "meta": {"id": "WUA_STAT_25K_008960", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=95.59, s=17.37, n=68, compute CI."}]}, "response": "95% CI ≈ [91.46, 99.72].", "meta": {"id": "WUA_STAT_25K_008961", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.42, compute E and Var."}]}, "response": "E[X]=13.02; Var[X]=7.552.", "meta": {"id": "WUA_STAT_25K_008962", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.18, compute E and Var."}]}, "response": "E[X]=2.88; Var[X]=2.362.", "meta": {"id": "WUA_STAT_25K_008963", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.88, s=11.74, n=139, compute CI."}]}, "response": "95% CI ≈ [170.93, 174.83].", "meta": {"id": "WUA_STAT_25K_008964", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.39, compute E and Var."}]}, "response": "E[X]=6.63; Var[X]=4.044.", "meta": {"id": "WUA_STAT_25K_008965", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.067, sens=0.76, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.168.", "meta": {"id": "WUA_STAT_25K_008966", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.89, s=39.55, n=49, compute CI."}]}, "response": "95% CI ≈ [103.82, 125.96].", "meta": {"id": "WUA_STAT_25K_008967", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.47, compute E and Var."}]}, "response": "E[X]=16.92; Var[X]=8.968.", "meta": {"id": "WUA_STAT_25K_008968", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.184, sens=0.96, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.591.", "meta": {"id": "WUA_STAT_25K_008969", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.043, sens=0.88, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.248.", "meta": {"id": "WUA_STAT_25K_008970", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.55, μ=11.51, σ=3.15, compute z."}]}, "response": "z = 13.663.", "meta": {"id": "WUA_STAT_25K_008971", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.195, sens=0.77, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.554.", "meta": {"id": "WUA_STAT_25K_008972", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.16, μ=71.1, σ=11.77, compute z."}]}, "response": "z = -2.714.", "meta": {"id": "WUA_STAT_25K_008973", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.61, μ=6.1, σ=8.64, compute z."}]}, "response": "z = 7.235.", "meta": {"id": "WUA_STAT_25K_008974", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.125, sens=0.89, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.459.", "meta": {"id": "WUA_STAT_25K_008975", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.31, μ=70.49, σ=20.83, compute z."}]}, "response": "z = -1.593.", "meta": {"id": "WUA_STAT_25K_008976", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.47, compute E and Var."}]}, "response": "E[X]=15.04; Var[X]=7.971.", "meta": {"id": "WUA_STAT_25K_008977", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.06, sens=0.98, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.363.", "meta": {"id": "WUA_STAT_25K_008978", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.14, compute E and Var."}]}, "response": "E[X]=1.68; Var[X]=1.445.", "meta": {"id": "WUA_STAT_25K_008979", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.79, compute E and Var."}]}, "response": "E[X]=12.64; Var[X]=2.654.", "meta": {"id": "WUA_STAT_25K_008980", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=195.35, s=31.28, n=101, compute CI."}]}, "response": "95% CI ≈ [189.25, 201.45].", "meta": {"id": "WUA_STAT_25K_008981", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.33, μ=79.63, σ=1.33, compute z."}]}, "response": "z = -25.038.", "meta": {"id": "WUA_STAT_25K_008982", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.77, compute E and Var."}]}, "response": "E[X]=20.02; Var[X]=4.605.", "meta": {"id": "WUA_STAT_25K_008983", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.69, μ=16.18, σ=6.61, compute z."}]}, "response": "z = 5.372.", "meta": {"id": "WUA_STAT_25K_008984", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.67, compute E and Var."}]}, "response": "E[X]=18.09; Var[X]=5.970.", "meta": {"id": "WUA_STAT_25K_008985", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.4, s=15.93, n=54, compute CI."}]}, "response": "95% CI ≈ [195.15, 203.65].", "meta": {"id": "WUA_STAT_25K_008986", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.38, s=37.45, n=25, compute CI."}]}, "response": "95% CI ≈ [59.70, 89.06].", "meta": {"id": "WUA_STAT_25K_008987", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.45, μ=41.85, σ=14.86, compute z."}]}, "response": "z = 1.184.", "meta": {"id": "WUA_STAT_25K_008988", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.76, s=13.01, n=85, compute CI."}]}, "response": "95% CI ≈ [196.99, 202.53].", "meta": {"id": "WUA_STAT_25K_008989", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=141.55, s=11.5, n=64, compute CI."}]}, "response": "95% CI ≈ [138.73, 144.37].", "meta": {"id": "WUA_STAT_25K_008990", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.089, sens=0.91, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.447.", "meta": {"id": "WUA_STAT_25K_008991", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.33, s=28.09, n=162, compute CI."}]}, "response": "95% CI ≈ [117.00, 125.66].", "meta": {"id": "WUA_STAT_25K_008992", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.39, compute E and Var."}]}, "response": "E[X]=14.82; Var[X]=9.040.", "meta": {"id": "WUA_STAT_25K_008993", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.95, μ=59.22, σ=23.36, compute z."}]}, "response": "z = -0.482.", "meta": {"id": "WUA_STAT_25K_008994", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.69, μ=41.97, σ=15.89, compute z."}]}, "response": "z = -2.094.", "meta": {"id": "WUA_STAT_25K_008995", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.26, μ=75.32, σ=1.93, compute z."}]}, "response": "z = -23.865.", "meta": {"id": "WUA_STAT_25K_008996", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.6, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=2.880.", "meta": {"id": "WUA_STAT_25K_008997", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.036, sens=0.9, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.324.", "meta": {"id": "WUA_STAT_25K_008998", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.02, μ=82.63, σ=18.36, compute z."}]}, "response": "z = -2.048.", "meta": {"id": "WUA_STAT_25K_008999", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.33, μ=81.0, σ=1.33, compute z."}]}, "response": "z = -29.827.", "meta": {"id": "WUA_STAT_25K_009000", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.085, sens=0.8, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.347.", "meta": {"id": "WUA_STAT_25K_009001", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.24, μ=1.25, σ=2.36, compute z."}]}, "response": "z = 22.877.", "meta": {"id": "WUA_STAT_25K_009002", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.09, μ=18.7, σ=5.48, compute z."}]}, "response": "z = 1.349.", "meta": {"id": "WUA_STAT_25K_009003", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.79, s=31.87, n=148, compute CI."}]}, "response": "95% CI ≈ [172.66, 182.92].", "meta": {"id": "WUA_STAT_25K_009004", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.06, sens=0.92, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.495.", "meta": {"id": "WUA_STAT_25K_009005", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.79, μ=22.83, σ=1.38, compute z."}]}, "response": "z = 53.594.", "meta": {"id": "WUA_STAT_25K_009006", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.031, sens=0.95, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.336.", "meta": {"id": "WUA_STAT_25K_009007", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.77, μ=93.45, σ=11.04, compute z."}]}, "response": "z = -2.236.", "meta": {"id": "WUA_STAT_25K_009008", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.36, μ=93.03, σ=6.25, compute z."}]}, "response": "z = -12.107.", "meta": {"id": "WUA_STAT_25K_009009", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.35, s=8.77, n=105, compute CI."}]}, "response": "95% CI ≈ [88.67, 92.03].", "meta": {"id": "WUA_STAT_25K_009010", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.95, s=25.54, n=194, compute CI."}]}, "response": "95% CI ≈ [50.36, 57.54].", "meta": {"id": "WUA_STAT_25K_009011", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.16, μ=26.47, σ=3.07, compute z."}]}, "response": "z = 1.202.", "meta": {"id": "WUA_STAT_25K_009012", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.24, s=13.45, n=22, compute CI."}]}, "response": "95% CI ≈ [115.62, 126.86].", "meta": {"id": "WUA_STAT_25K_009013", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.16, compute E and Var."}]}, "response": "E[X]=5.92; Var[X]=4.973.", "meta": {"id": "WUA_STAT_25K_009014", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.086, sens=0.89, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.267.", "meta": {"id": "WUA_STAT_25K_009015", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.1, μ=23.59, σ=16.38, compute z."}]}, "response": "z = -0.885.", "meta": {"id": "WUA_STAT_25K_009016", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.87, μ=88.12, σ=17.09, compute z."}]}, "response": "z = -1.653.", "meta": {"id": "WUA_STAT_25K_009017", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.06, μ=44.46, σ=12.2, compute z."}]}, "response": "z = 3.820.", "meta": {"id": "WUA_STAT_25K_009018", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.014, sens=0.92, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.056.", "meta": {"id": "WUA_STAT_25K_009019", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.1, s=31.15, n=190, compute CI."}]}, "response": "95% CI ≈ [140.67, 149.53].", "meta": {"id": "WUA_STAT_25K_009020", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.55, μ=85.26, σ=10.94, compute z."}]}, "response": "z = -6.829.", "meta": {"id": "WUA_STAT_25K_009021", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.11, μ=27.44, σ=4.64, compute z."}]}, "response": "z = 12.213.", "meta": {"id": "WUA_STAT_25K_009022", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.1, compute E and Var."}]}, "response": "E[X]=4.00; Var[X]=3.600.", "meta": {"id": "WUA_STAT_25K_009023", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.34, μ=73.26, σ=12.92, compute z."}]}, "response": "z = -5.025.", "meta": {"id": "WUA_STAT_25K_009024", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.86, s=3.13, n=110, compute CI."}]}, "response": "95% CI ≈ [55.28, 56.44].", "meta": {"id": "WUA_STAT_25K_009025", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.24, s=29.4, n=111, compute CI."}]}, "response": "95% CI ≈ [80.77, 91.71].", "meta": {"id": "WUA_STAT_25K_009026", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.3, s=19.56, n=123, compute CI."}]}, "response": "95% CI ≈ [115.84, 122.76].", "meta": {"id": "WUA_STAT_25K_009027", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.059, sens=0.99, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.437.", "meta": {"id": "WUA_STAT_25K_009028", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.91, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.676.", "meta": {"id": "WUA_STAT_25K_009029", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=72.9, s=1.92, n=159, compute CI."}]}, "response": "95% CI ≈ [72.60, 73.20].", "meta": {"id": "WUA_STAT_25K_009030", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.89, μ=56.79, σ=17.62, compute z."}]}, "response": "z = -2.889.", "meta": {"id": "WUA_STAT_25K_009031", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.6, compute E and Var."}]}, "response": "E[X]=19.20; Var[X]=7.680.", "meta": {"id": "WUA_STAT_25K_009032", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.64, compute E and Var."}]}, "response": "E[X]=12.80; Var[X]=4.608.", "meta": {"id": "WUA_STAT_25K_009033", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.92, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.293.", "meta": {"id": "WUA_STAT_25K_009034", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.231, sens=0.94, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.530.", "meta": {"id": "WUA_STAT_25K_009035", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.52, μ=20.01, σ=12.74, compute z."}]}, "response": "z = -0.117.", "meta": {"id": "WUA_STAT_25K_009036", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.06, compute E and Var."}]}, "response": "E[X]=1.56; Var[X]=1.466.", "meta": {"id": "WUA_STAT_25K_009037", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.09, compute E and Var."}]}, "response": "E[X]=1.53; Var[X]=1.392.", "meta": {"id": "WUA_STAT_25K_009038", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.084, sens=0.82, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.485.", "meta": {"id": "WUA_STAT_25K_009039", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.96, s=5.31, n=190, compute CI."}]}, "response": "95% CI ≈ [107.20, 108.72].", "meta": {"id": "WUA_STAT_25K_009040", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.26, s=21.73, n=70, compute CI."}]}, "response": "95% CI ≈ [158.17, 168.35].", "meta": {"id": "WUA_STAT_25K_009041", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.14, s=10.25, n=165, compute CI."}]}, "response": "95% CI ≈ [84.58, 87.70].", "meta": {"id": "WUA_STAT_25K_009042", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.86, μ=59.73, σ=1.57, compute z."}]}, "response": "z = 25.561.", "meta": {"id": "WUA_STAT_25K_009043", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.26, μ=66.67, σ=21.08, compute z."}]}, "response": "z = -0.921.", "meta": {"id": "WUA_STAT_25K_009044", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.01, μ=74.69, σ=22.45, compute z."}]}, "response": "z = 0.816.", "meta": {"id": "WUA_STAT_25K_009045", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.97, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.638.", "meta": {"id": "WUA_STAT_25K_009046", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.08, compute E and Var."}]}, "response": "E[X]=1.52; Var[X]=1.398.", "meta": {"id": "WUA_STAT_25K_009047", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.23, μ=57.45, σ=6.2, compute z."}]}, "response": "z = -2.616.", "meta": {"id": "WUA_STAT_25K_009048", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.47, μ=97.54, σ=7.32, compute z."}]}, "response": "z = -6.430.", "meta": {"id": "WUA_STAT_25K_009049", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.82, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.297.", "meta": {"id": "WUA_STAT_25K_009050", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=115.08, s=17.05, n=154, compute CI."}]}, "response": "95% CI ≈ [112.39, 117.77].", "meta": {"id": "WUA_STAT_25K_009051", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.44, compute E and Var."}]}, "response": "E[X]=9.24; Var[X]=5.174.", "meta": {"id": "WUA_STAT_25K_009052", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.133, sens=0.84, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.392.", "meta": {"id": "WUA_STAT_25K_009053", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.052, sens=0.89, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.163.", "meta": {"id": "WUA_STAT_25K_009054", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.217, sens=0.93, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.479.", "meta": {"id": "WUA_STAT_25K_009055", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.86, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.233.", "meta": {"id": "WUA_STAT_25K_009056", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.06, μ=94.85, σ=1.45, compute z."}]}, "response": "z = -51.579.", "meta": {"id": "WUA_STAT_25K_009057", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.26, μ=96.66, σ=5.61, compute z."}]}, "response": "z = -8.806.", "meta": {"id": "WUA_STAT_25K_009058", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.36, compute E and Var."}]}, "response": "E[X]=11.16; Var[X]=7.142.", "meta": {"id": "WUA_STAT_25K_009059", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.68, s=20.16, n=120, compute CI."}]}, "response": "95% CI ≈ [163.07, 170.29].", "meta": {"id": "WUA_STAT_25K_009060", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.31, s=16.98, n=109, compute CI."}]}, "response": "95% CI ≈ [133.12, 139.50].", "meta": {"id": "WUA_STAT_25K_009061", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.7, s=30.81, n=18, compute CI."}]}, "response": "95% CI ≈ [27.47, 55.93].", "meta": {"id": "WUA_STAT_25K_009062", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.43, compute E and Var."}]}, "response": "E[X]=17.20; Var[X]=9.804.", "meta": {"id": "WUA_STAT_25K_009063", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.65, compute E and Var."}]}, "response": "E[X]=21.45; Var[X]=7.507.", "meta": {"id": "WUA_STAT_25K_009064", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.224, sens=0.94, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.844.", "meta": {"id": "WUA_STAT_25K_009065", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.84, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.293.", "meta": {"id": "WUA_STAT_25K_009066", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.092, sens=0.93, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.461.", "meta": {"id": "WUA_STAT_25K_009067", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.69, μ=88.31, σ=14.92, compute z."}]}, "response": "z = -5.471.", "meta": {"id": "WUA_STAT_25K_009068", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.5, μ=83.97, σ=10.81, compute z."}]}, "response": "z = -4.021.", "meta": {"id": "WUA_STAT_25K_009069", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.111, sens=0.86, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.349.", "meta": {"id": "WUA_STAT_25K_009070", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.77, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.588.", "meta": {"id": "WUA_STAT_25K_009071", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.27, s=29.91, n=50, compute CI."}]}, "response": "95% CI ≈ [158.98, 175.56].", "meta": {"id": "WUA_STAT_25K_009072", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.054, sens=0.88, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.264.", "meta": {"id": "WUA_STAT_25K_009073", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.51, s=13.92, n=90, compute CI."}]}, "response": "95% CI ≈ [119.63, 125.39].", "meta": {"id": "WUA_STAT_25K_009074", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.69, μ=50.11, σ=18.79, compute z."}]}, "response": "z = -1.406.", "meta": {"id": "WUA_STAT_25K_009075", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.24, sens=0.79, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.462.", "meta": {"id": "WUA_STAT_25K_009076", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.34, μ=85.79, σ=7.82, compute z."}]}, "response": "z = -4.022.", "meta": {"id": "WUA_STAT_25K_009077", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.19, μ=29.86, σ=4.2, compute z."}]}, "response": "z = 5.317.", "meta": {"id": "WUA_STAT_25K_009078", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.27, s=36.37, n=131, compute CI."}]}, "response": "95% CI ≈ [145.04, 157.50].", "meta": {"id": "WUA_STAT_25K_009079", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.32, μ=43.35, σ=15.82, compute z."}]}, "response": "z = 0.377.", "meta": {"id": "WUA_STAT_25K_009080", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.08, s=4.55, n=181, compute CI."}]}, "response": "95% CI ≈ [62.42, 63.74].", "meta": {"id": "WUA_STAT_25K_009081", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.7, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.350.", "meta": {"id": "WUA_STAT_25K_009082", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.56, μ=93.78, σ=7.84, compute z."}]}, "response": "z = -2.324.", "meta": {"id": "WUA_STAT_25K_009083", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.91, μ=5.4, σ=24.8, compute z."}]}, "response": "z = 3.327.", "meta": {"id": "WUA_STAT_25K_009084", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.74, s=15.53, n=36, compute CI."}]}, "response": "95% CI ≈ [78.67, 88.81].", "meta": {"id": "WUA_STAT_25K_009085", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.53, μ=42.26, σ=10.03, compute z."}]}, "response": "z = -2.665.", "meta": {"id": "WUA_STAT_25K_009086", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.17, μ=43.22, σ=8.37, compute z."}]}, "response": "z = 4.176.", "meta": {"id": "WUA_STAT_25K_009087", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.3, μ=33.09, σ=17.11, compute z."}]}, "response": "z = -0.222.", "meta": {"id": "WUA_STAT_25K_009088", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.46, μ=74.83, σ=4.49, compute z."}]}, "response": "z = 5.263.", "meta": {"id": "WUA_STAT_25K_009089", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.25, compute E and Var."}]}, "response": "E[X]=7.25; Var[X]=5.438.", "meta": {"id": "WUA_STAT_25K_009090", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.249, sens=0.99, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.531.", "meta": {"id": "WUA_STAT_25K_009091", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.058, sens=0.83, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.836.", "meta": {"id": "WUA_STAT_25K_009092", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.84, μ=58.75, σ=11.73, compute z."}]}, "response": "z = 3.418.", "meta": {"id": "WUA_STAT_25K_009093", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.95, s=32.63, n=73, compute CI."}]}, "response": "95% CI ≈ [53.46, 68.44].", "meta": {"id": "WUA_STAT_25K_009094", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.71, μ=60.84, σ=24.44, compute z."}]}, "response": "z = 0.158.", "meta": {"id": "WUA_STAT_25K_009095", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.34, s=17.63, n=17, compute CI."}]}, "response": "95% CI ≈ [48.96, 65.72].", "meta": {"id": "WUA_STAT_25K_009096", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.4, compute E and Var."}]}, "response": "E[X]=12.00; Var[X]=7.200.", "meta": {"id": "WUA_STAT_25K_009097", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.1, compute E and Var."}]}, "response": "E[X]=1.90; Var[X]=1.710.", "meta": {"id": "WUA_STAT_25K_009098", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.14, s=17.37, n=166, compute CI."}]}, "response": "95% CI ≈ [41.50, 46.78].", "meta": {"id": "WUA_STAT_25K_009099", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.96, s=6.93, n=80, compute CI."}]}, "response": "95% CI ≈ [178.44, 181.48].", "meta": {"id": "WUA_STAT_25K_009100", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.169, sens=0.98, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.666.", "meta": {"id": "WUA_STAT_25K_009101", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.66, compute E and Var."}]}, "response": "E[X]=7.26; Var[X]=2.468.", "meta": {"id": "WUA_STAT_25K_009102", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=94.86, s=30.82, n=136, compute CI."}]}, "response": "95% CI ≈ [89.68, 100.04].", "meta": {"id": "WUA_STAT_25K_009103", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.52, compute E and Var."}]}, "response": "E[X]=3.12; Var[X]=1.498.", "meta": {"id": "WUA_STAT_25K_009104", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.58, compute E and Var."}]}, "response": "E[X]=5.80; Var[X]=2.436.", "meta": {"id": "WUA_STAT_25K_009105", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.84, s=30.39, n=94, compute CI."}]}, "response": "95% CI ≈ [119.70, 131.98].", "meta": {"id": "WUA_STAT_25K_009106", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.69, μ=97.23, σ=20.06, compute z."}]}, "response": "z = -3.167.", "meta": {"id": "WUA_STAT_25K_009107", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.41, μ=41.08, σ=18.38, compute z."}]}, "response": "z = -0.526.", "meta": {"id": "WUA_STAT_25K_009108", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.99, μ=54.64, σ=4.09, compute z."}]}, "response": "z = -1.626.", "meta": {"id": "WUA_STAT_25K_009109", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=108.61, s=38.87, n=191, compute CI."}]}, "response": "95% CI ≈ [103.10, 114.12].", "meta": {"id": "WUA_STAT_25K_009110", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=94.02, s=21.37, n=82, compute CI."}]}, "response": "95% CI ≈ [89.39, 98.65].", "meta": {"id": "WUA_STAT_25K_009111", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.15, s=6.14, n=103, compute CI."}]}, "response": "95% CI ≈ [154.96, 157.34].", "meta": {"id": "WUA_STAT_25K_009112", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.7, compute E and Var."}]}, "response": "E[X]=7.00; Var[X]=2.100.", "meta": {"id": "WUA_STAT_25K_009113", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.88, s=25.52, n=195, compute CI."}]}, "response": "95% CI ≈ [10.30, 17.46].", "meta": {"id": "WUA_STAT_25K_009114", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.2, s=34.16, n=181, compute CI."}]}, "response": "95% CI ≈ [162.22, 172.18].", "meta": {"id": "WUA_STAT_25K_009115", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.97, μ=57.21, σ=8.71, compute z."}]}, "response": "z = -4.161.", "meta": {"id": "WUA_STAT_25K_009116", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.71, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.495.", "meta": {"id": "WUA_STAT_25K_009117", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.027, sens=0.76, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.161.", "meta": {"id": "WUA_STAT_25K_009118", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=28.72, s=20.54, n=152, compute CI."}]}, "response": "95% CI ≈ [25.45, 31.99].", "meta": {"id": "WUA_STAT_25K_009119", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.91, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.802.", "meta": {"id": "WUA_STAT_25K_009120", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.182, sens=0.77, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.388.", "meta": {"id": "WUA_STAT_25K_009121", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.234, sens=0.76, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.453.", "meta": {"id": "WUA_STAT_25K_009122", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.157, sens=0.84, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.887.", "meta": {"id": "WUA_STAT_25K_009123", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.5, compute E and Var."}]}, "response": "E[X]=10.00; Var[X]=5.000.", "meta": {"id": "WUA_STAT_25K_009124", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.37, s=38.49, n=137, compute CI."}]}, "response": "95% CI ≈ [43.92, 56.82].", "meta": {"id": "WUA_STAT_25K_009125", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.05, μ=89.92, σ=17.59, compute z."}]}, "response": "z = -4.200.", "meta": {"id": "WUA_STAT_25K_009126", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.84, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.689.", "meta": {"id": "WUA_STAT_25K_009127", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.4, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=4.320.", "meta": {"id": "WUA_STAT_25K_009128", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.66, s=28.93, n=124, compute CI."}]}, "response": "95% CI ≈ [54.57, 64.75].", "meta": {"id": "WUA_STAT_25K_009129", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.95, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.403.", "meta": {"id": "WUA_STAT_25K_009130", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.69, μ=32.69, σ=11.21, compute z."}]}, "response": "z = -1.070.", "meta": {"id": "WUA_STAT_25K_009131", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.014, sens=0.82, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.050.", "meta": {"id": "WUA_STAT_25K_009132", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.34, compute E and Var."}]}, "response": "E[X]=4.76; Var[X]=3.142.", "meta": {"id": "WUA_STAT_25K_009133", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.21, compute E and Var."}]}, "response": "E[X]=3.78; Var[X]=2.986.", "meta": {"id": "WUA_STAT_25K_009134", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.44, s=8.31, n=117, compute CI."}]}, "response": "95% CI ≈ [115.93, 118.95].", "meta": {"id": "WUA_STAT_25K_009135", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.02, s=6.59, n=54, compute CI."}]}, "response": "95% CI ≈ [65.26, 68.78].", "meta": {"id": "WUA_STAT_25K_009136", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.7, s=10.39, n=70, compute CI."}]}, "response": "95% CI ≈ [116.27, 121.13].", "meta": {"id": "WUA_STAT_25K_009137", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.125, sens=0.91, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.325.", "meta": {"id": "WUA_STAT_25K_009138", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.71, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.288.", "meta": {"id": "WUA_STAT_25K_009139", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.4, compute E and Var."}]}, "response": "E[X]=7.60; Var[X]=4.560.", "meta": {"id": "WUA_STAT_25K_009140", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.51, μ=85.84, σ=9.48, compute z."}]}, "response": "z = 0.493.", "meta": {"id": "WUA_STAT_25K_009141", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.34, s=5.25, n=77, compute CI."}]}, "response": "95% CI ≈ [137.17, 139.51].", "meta": {"id": "WUA_STAT_25K_009142", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.63, s=27.79, n=108, compute CI."}]}, "response": "95% CI ≈ [75.39, 85.87].", "meta": {"id": "WUA_STAT_25K_009143", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.85, s=1.02, n=64, compute CI."}]}, "response": "95% CI ≈ [106.60, 107.10].", "meta": {"id": "WUA_STAT_25K_009144", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.55, μ=37.39, σ=2.25, compute z."}]}, "response": "z = 0.960.", "meta": {"id": "WUA_STAT_25K_009145", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.4, μ=23.18, σ=21.29, compute z."}]}, "response": "z = 1.607.", "meta": {"id": "WUA_STAT_25K_009146", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.35, μ=78.02, σ=23.17, compute z."}]}, "response": "z = -2.662.", "meta": {"id": "WUA_STAT_25K_009147", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.94, s=24.84, n=131, compute CI."}]}, "response": "95% CI ≈ [20.69, 29.19].", "meta": {"id": "WUA_STAT_25K_009148", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.12, μ=81.38, σ=24.33, compute z."}]}, "response": "z = -1.573.", "meta": {"id": "WUA_STAT_25K_009149", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.7, compute E and Var."}]}, "response": "E[X]=21.00; Var[X]=6.300.", "meta": {"id": "WUA_STAT_25K_009150", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.6, compute E and Var."}]}, "response": "E[X]=13.20; Var[X]=5.280.", "meta": {"id": "WUA_STAT_25K_009151", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.92, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.581.", "meta": {"id": "WUA_STAT_25K_009152", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.76, μ=31.92, σ=4.88, compute z."}]}, "response": "z = 1.197.", "meta": {"id": "WUA_STAT_25K_009153", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.36, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=1.613.", "meta": {"id": "WUA_STAT_25K_009154", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.96, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.322.", "meta": {"id": "WUA_STAT_25K_009155", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.17, s=1.78, n=157, compute CI."}]}, "response": "95% CI ≈ [174.89, 175.45].", "meta": {"id": "WUA_STAT_25K_009156", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.64, compute E and Var."}]}, "response": "E[X]=14.72; Var[X]=5.299.", "meta": {"id": "WUA_STAT_25K_009157", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.63, s=12.73, n=71, compute CI."}]}, "response": "95% CI ≈ [173.67, 179.59].", "meta": {"id": "WUA_STAT_25K_009158", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.94, μ=78.47, σ=23.94, compute z."}]}, "response": "z = 0.020.", "meta": {"id": "WUA_STAT_25K_009159", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.97, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.371.", "meta": {"id": "WUA_STAT_25K_009160", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.143, sens=0.77, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.588.", "meta": {"id": "WUA_STAT_25K_009161", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.167, sens=0.96, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.681.", "meta": {"id": "WUA_STAT_25K_009162", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.143, sens=0.93, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.414.", "meta": {"id": "WUA_STAT_25K_009163", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.186, sens=0.82, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.524.", "meta": {"id": "WUA_STAT_25K_009164", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.72, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.324.", "meta": {"id": "WUA_STAT_25K_009165", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.28, compute E and Var."}]}, "response": "E[X]=1.68; Var[X]=1.210.", "meta": {"id": "WUA_STAT_25K_009166", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.72, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.458.", "meta": {"id": "WUA_STAT_25K_009167", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.71, compute E and Var."}]}, "response": "E[X]=22.72; Var[X]=6.589.", "meta": {"id": "WUA_STAT_25K_009168", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.06, compute E and Var."}]}, "response": "E[X]=0.72; Var[X]=0.677.", "meta": {"id": "WUA_STAT_25K_009169", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.62, compute E and Var."}]}, "response": "E[X]=20.46; Var[X]=7.775.", "meta": {"id": "WUA_STAT_25K_009170", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.119, sens=0.72, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.364.", "meta": {"id": "WUA_STAT_25K_009171", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.066, sens=0.93, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.267.", "meta": {"id": "WUA_STAT_25K_009172", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.02, s=5.59, n=178, compute CI."}]}, "response": "95% CI ≈ [158.20, 159.84].", "meta": {"id": "WUA_STAT_25K_009173", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.95, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.423.", "meta": {"id": "WUA_STAT_25K_009174", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.34, μ=81.93, σ=10.49, compute z."}]}, "response": "z = 0.134.", "meta": {"id": "WUA_STAT_25K_009175", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.36, s=17.64, n=108, compute CI."}]}, "response": "95% CI ≈ [27.03, 33.69].", "meta": {"id": "WUA_STAT_25K_009176", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.92, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.596.", "meta": {"id": "WUA_STAT_25K_009177", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.8, compute E and Var."}]}, "response": "E[X]=22.40; Var[X]=4.480.", "meta": {"id": "WUA_STAT_25K_009178", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.26, μ=27.97, σ=4.29, compute z."}]}, "response": "z = 2.166.", "meta": {"id": "WUA_STAT_25K_009179", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.77, μ=68.62, σ=9.87, compute z."}]}, "response": "z = 2.042.", "meta": {"id": "WUA_STAT_25K_009180", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.52, compute E and Var."}]}, "response": "E[X]=7.80; Var[X]=3.744.", "meta": {"id": "WUA_STAT_25K_009181", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.1, μ=70.05, σ=4.49, compute z."}]}, "response": "z = -11.793.", "meta": {"id": "WUA_STAT_25K_009182", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.71, compute E and Var."}]}, "response": "E[X]=24.14; Var[X]=7.001.", "meta": {"id": "WUA_STAT_25K_009183", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.7, s=8.66, n=170, compute CI."}]}, "response": "95% CI ≈ [59.40, 62.00].", "meta": {"id": "WUA_STAT_25K_009184", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.59, compute E and Var."}]}, "response": "E[X]=17.11; Var[X]=7.015.", "meta": {"id": "WUA_STAT_25K_009185", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.74, compute E and Var."}]}, "response": "E[X]=11.10; Var[X]=2.886.", "meta": {"id": "WUA_STAT_25K_009186", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.69, compute E and Var."}]}, "response": "E[X]=24.15; Var[X]=7.487.", "meta": {"id": "WUA_STAT_25K_009187", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.93, μ=78.3, σ=21.44, compute z."}]}, "response": "z = 0.216.", "meta": {"id": "WUA_STAT_25K_009188", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.68, μ=9.02, σ=13.73, compute z."}]}, "response": "z = 2.087.", "meta": {"id": "WUA_STAT_25K_009189", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.35, μ=96.52, σ=1.94, compute z."}]}, "response": "z = -40.294.", "meta": {"id": "WUA_STAT_25K_009190", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.33, μ=40.61, σ=6.75, compute z."}]}, "response": "z = -2.856.", "meta": {"id": "WUA_STAT_25K_009191", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.046, sens=0.98, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.612.", "meta": {"id": "WUA_STAT_25K_009192", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.42, μ=1.51, σ=3.85, compute z."}]}, "response": "z = 17.899.", "meta": {"id": "WUA_STAT_25K_009193", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.09, compute E and Var."}]}, "response": "E[X]=1.62; Var[X]=1.474.", "meta": {"id": "WUA_STAT_25K_009194", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.18, s=1.54, n=136, compute CI."}]}, "response": "95% CI ≈ [91.92, 92.44].", "meta": {"id": "WUA_STAT_25K_009195", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.68, μ=87.28, σ=11.09, compute z."}]}, "response": "z = -7.358.", "meta": {"id": "WUA_STAT_25K_009196", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.39, compute E and Var."}]}, "response": "E[X]=10.92; Var[X]=6.661.", "meta": {"id": "WUA_STAT_25K_009197", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.95, s=19.18, n=74, compute CI."}]}, "response": "95% CI ≈ [195.58, 204.32].", "meta": {"id": "WUA_STAT_25K_009198", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.46, compute E and Var."}]}, "response": "E[X]=2.30; Var[X]=1.242.", "meta": {"id": "WUA_STAT_25K_009199", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.25, sens=0.89, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.533.", "meta": {"id": "WUA_STAT_25K_009200", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.71, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.155.", "meta": {"id": "WUA_STAT_25K_009201", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.38, μ=33.04, σ=8.03, compute z."}]}, "response": "z = 2.533.", "meta": {"id": "WUA_STAT_25K_009202", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.137, sens=0.75, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.323.", "meta": {"id": "WUA_STAT_25K_009203", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.52, compute E and Var."}]}, "response": "E[X]=19.76; Var[X]=9.485.", "meta": {"id": "WUA_STAT_25K_009204", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.44, μ=82.72, σ=11.21, compute z."}]}, "response": "z = -5.467.", "meta": {"id": "WUA_STAT_25K_009205", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.18, s=8.41, n=28, compute CI."}]}, "response": "95% CI ≈ [55.06, 61.30].", "meta": {"id": "WUA_STAT_25K_009206", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.65, s=1.74, n=196, compute CI."}]}, "response": "95% CI ≈ [79.41, 79.89].", "meta": {"id": "WUA_STAT_25K_009207", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.88, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.672.", "meta": {"id": "WUA_STAT_25K_009208", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.029, sens=0.71, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.298.", "meta": {"id": "WUA_STAT_25K_009209", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.9, s=31.53, n=45, compute CI."}]}, "response": "95% CI ≈ [175.69, 194.11].", "meta": {"id": "WUA_STAT_25K_009210", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.01, s=11.62, n=102, compute CI."}]}, "response": "95% CI ≈ [141.75, 146.27].", "meta": {"id": "WUA_STAT_25K_009211", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.25, s=26.66, n=19, compute CI."}]}, "response": "95% CI ≈ [144.26, 168.24].", "meta": {"id": "WUA_STAT_25K_009212", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=108.23, s=22.45, n=178, compute CI."}]}, "response": "95% CI ≈ [104.93, 111.53].", "meta": {"id": "WUA_STAT_25K_009213", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.58, μ=67.23, σ=16.32, compute z."}]}, "response": "z = 1.553.", "meta": {"id": "WUA_STAT_25K_009214", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.94, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.898.", "meta": {"id": "WUA_STAT_25K_009215", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.99, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.192.", "meta": {"id": "WUA_STAT_25K_009216", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.91, μ=56.32, σ=19.51, compute z."}]}, "response": "z = -1.251.", "meta": {"id": "WUA_STAT_25K_009217", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.56, μ=60.93, σ=23.39, compute z."}]}, "response": "z = -1.341.", "meta": {"id": "WUA_STAT_25K_009218", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.24, μ=48.11, σ=23.51, compute z."}]}, "response": "z = -1.228.", "meta": {"id": "WUA_STAT_25K_009219", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.19, compute E and Var."}]}, "response": "E[X]=6.65; Var[X]=5.387.", "meta": {"id": "WUA_STAT_25K_009220", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.59, compute E and Var."}]}, "response": "E[X]=20.65; Var[X]=8.466.", "meta": {"id": "WUA_STAT_25K_009221", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.87, s=4.04, n=191, compute CI."}]}, "response": "95% CI ≈ [60.30, 61.44].", "meta": {"id": "WUA_STAT_25K_009222", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.18, μ=86.57, σ=11.68, compute z."}]}, "response": "z = -5.170.", "meta": {"id": "WUA_STAT_25K_009223", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=105.15, s=11.96, n=113, compute CI."}]}, "response": "95% CI ≈ [102.94, 107.36].", "meta": {"id": "WUA_STAT_25K_009224", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.09, μ=79.48, σ=7.73, compute z."}]}, "response": "z = -5.225.", "meta": {"id": "WUA_STAT_25K_009225", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.066, sens=0.8, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.320.", "meta": {"id": "WUA_STAT_25K_009226", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.63, compute E and Var."}]}, "response": "E[X]=4.41; Var[X]=1.632.", "meta": {"id": "WUA_STAT_25K_009227", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.234, sens=0.97, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.622.", "meta": {"id": "WUA_STAT_25K_009228", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.208, sens=0.79, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.485.", "meta": {"id": "WUA_STAT_25K_009229", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.12, s=30.39, n=33, compute CI."}]}, "response": "95% CI ≈ [103.75, 124.49].", "meta": {"id": "WUA_STAT_25K_009230", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.96, s=7.72, n=36, compute CI."}]}, "response": "95% CI ≈ [46.44, 51.48].", "meta": {"id": "WUA_STAT_25K_009231", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.3, μ=5.7, σ=17.09, compute z."}]}, "response": "z = 3.487.", "meta": {"id": "WUA_STAT_25K_009232", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.117, sens=0.83, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.282.", "meta": {"id": "WUA_STAT_25K_009233", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.43, compute E and Var."}]}, "response": "E[X]=7.74; Var[X]=4.412.", "meta": {"id": "WUA_STAT_25K_009234", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.43, s=21.88, n=71, compute CI."}]}, "response": "95% CI ≈ [147.34, 157.52].", "meta": {"id": "WUA_STAT_25K_009235", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.03, μ=45.74, σ=3.04, compute z."}]}, "response": "z = -6.812.", "meta": {"id": "WUA_STAT_25K_009236", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.36, compute E and Var."}]}, "response": "E[X]=9.72; Var[X]=6.221.", "meta": {"id": "WUA_STAT_25K_009237", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.72, s=5.37, n=52, compute CI."}]}, "response": "95% CI ≈ [136.26, 139.18].", "meta": {"id": "WUA_STAT_25K_009238", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.11, μ=74.23, σ=4.14, compute z."}]}, "response": "z = 0.213.", "meta": {"id": "WUA_STAT_25K_009239", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.95, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.578.", "meta": {"id": "WUA_STAT_25K_009240", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=66.12, s=24.58, n=53, compute CI."}]}, "response": "95% CI ≈ [59.50, 72.74].", "meta": {"id": "WUA_STAT_25K_009241", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.1, s=11.42, n=112, compute CI."}]}, "response": "95% CI ≈ [55.98, 60.22].", "meta": {"id": "WUA_STAT_25K_009242", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.171, sens=0.79, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.597.", "meta": {"id": "WUA_STAT_25K_009243", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.29, s=2.05, n=24, compute CI."}]}, "response": "95% CI ≈ [61.47, 63.11].", "meta": {"id": "WUA_STAT_25K_009244", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.62, μ=74.67, σ=2.16, compute z."}]}, "response": "z = -13.912.", "meta": {"id": "WUA_STAT_25K_009245", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.41, compute E and Var."}]}, "response": "E[X]=6.97; Var[X]=4.112.", "meta": {"id": "WUA_STAT_25K_009246", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.03, μ=73.74, σ=9.14, compute z."}]}, "response": "z = 2.439.", "meta": {"id": "WUA_STAT_25K_009247", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.06, compute E and Var."}]}, "response": "E[X]=1.26; Var[X]=1.184.", "meta": {"id": "WUA_STAT_25K_009248", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.62, s=23.23, n=128, compute CI."}]}, "response": "95% CI ≈ [175.60, 183.64].", "meta": {"id": "WUA_STAT_25K_009249", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.19, μ=24.35, σ=17.55, compute z."}]}, "response": "z = 3.809.", "meta": {"id": "WUA_STAT_25K_009250", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.65, μ=83.77, σ=5.67, compute z."}]}, "response": "z = -3.901.", "meta": {"id": "WUA_STAT_25K_009251", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.9, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.579.", "meta": {"id": "WUA_STAT_25K_009252", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.07, μ=75.6, σ=10.96, compute z."}]}, "response": "z = -2.694.", "meta": {"id": "WUA_STAT_25K_009253", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.97, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.674.", "meta": {"id": "WUA_STAT_25K_009254", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.51, compute E and Var."}]}, "response": "E[X]=6.63; Var[X]=3.249.", "meta": {"id": "WUA_STAT_25K_009255", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.6, s=38.74, n=75, compute CI."}]}, "response": "95% CI ≈ [15.83, 33.37].", "meta": {"id": "WUA_STAT_25K_009256", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.43, μ=33.9, σ=2.66, compute z."}]}, "response": "z = 12.605.", "meta": {"id": "WUA_STAT_25K_009257", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.192, sens=0.91, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.464.", "meta": {"id": "WUA_STAT_25K_009258", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.58, s=12.19, n=26, compute CI."}]}, "response": "95% CI ≈ [102.89, 112.27].", "meta": {"id": "WUA_STAT_25K_009259", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.61, s=3.26, n=80, compute CI."}]}, "response": "95% CI ≈ [145.90, 147.32].", "meta": {"id": "WUA_STAT_25K_009260", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.73, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.617.", "meta": {"id": "WUA_STAT_25K_009261", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.57, compute E and Var."}]}, "response": "E[X]=7.41; Var[X]=3.186.", "meta": {"id": "WUA_STAT_25K_009262", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.87, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.757.", "meta": {"id": "WUA_STAT_25K_009263", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.1, μ=91.95, σ=2.6, compute z."}]}, "response": "z = -5.327.", "meta": {"id": "WUA_STAT_25K_009264", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.91, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.353.", "meta": {"id": "WUA_STAT_25K_009265", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.28, compute E and Var."}]}, "response": "E[X]=10.36; Var[X]=7.459.", "meta": {"id": "WUA_STAT_25K_009266", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.84, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.485.", "meta": {"id": "WUA_STAT_25K_009267", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.59, s=12.2, n=25, compute CI."}]}, "response": "95% CI ≈ [18.81, 28.37].", "meta": {"id": "WUA_STAT_25K_009268", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.19, compute E and Var."}]}, "response": "E[X]=7.41; Var[X]=6.002.", "meta": {"id": "WUA_STAT_25K_009269", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.77, s=25.65, n=94, compute CI."}]}, "response": "95% CI ≈ [82.58, 92.96].", "meta": {"id": "WUA_STAT_25K_009270", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.0, μ=92.67, σ=21.15, compute z."}]}, "response": "z = -2.443.", "meta": {"id": "WUA_STAT_25K_009271", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.41, μ=19.75, σ=1.11, compute z."}]}, "response": "z = 60.955.", "meta": {"id": "WUA_STAT_25K_009272", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.82, μ=99.27, σ=7.75, compute z."}]}, "response": "z = -11.800.", "meta": {"id": "WUA_STAT_25K_009273", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.97, s=12.61, n=168, compute CI."}]}, "response": "95% CI ≈ [192.06, 195.88].", "meta": {"id": "WUA_STAT_25K_009274", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.9, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.162.", "meta": {"id": "WUA_STAT_25K_009275", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.7, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.073.", "meta": {"id": "WUA_STAT_25K_009276", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.11, s=33.33, n=51, compute CI."}]}, "response": "95% CI ≈ [171.96, 190.26].", "meta": {"id": "WUA_STAT_25K_009277", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.74, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.240.", "meta": {"id": "WUA_STAT_25K_009278", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.53, compute E and Var."}]}, "response": "E[X]=18.55; Var[X]=8.719.", "meta": {"id": "WUA_STAT_25K_009279", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.61, s=28.02, n=108, compute CI."}]}, "response": "95% CI ≈ [56.33, 66.89].", "meta": {"id": "WUA_STAT_25K_009280", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.49, s=27.25, n=46, compute CI."}]}, "response": "95% CI ≈ [42.62, 58.36].", "meta": {"id": "WUA_STAT_25K_009281", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.42, μ=8.67, σ=13.13, compute z."}]}, "response": "z = 5.769.", "meta": {"id": "WUA_STAT_25K_009282", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.79, compute E and Var."}]}, "response": "E[X]=9.48; Var[X]=1.991.", "meta": {"id": "WUA_STAT_25K_009283", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.103, sens=0.9, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.775.", "meta": {"id": "WUA_STAT_25K_009284", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.108, sens=0.96, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.346.", "meta": {"id": "WUA_STAT_25K_009285", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.23, compute E and Var."}]}, "response": "E[X]=8.51; Var[X]=6.553.", "meta": {"id": "WUA_STAT_25K_009286", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.06, μ=32.32, σ=6.11, compute z."}]}, "response": "z = 2.903.", "meta": {"id": "WUA_STAT_25K_009287", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.67, compute E and Var."}]}, "response": "E[X]=10.05; Var[X]=3.317.", "meta": {"id": "WUA_STAT_25K_009288", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.223, sens=0.7, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.691.", "meta": {"id": "WUA_STAT_25K_009289", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=168.19, s=20.09, n=110, compute CI."}]}, "response": "95% CI ≈ [164.44, 171.94].", "meta": {"id": "WUA_STAT_25K_009290", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.14, compute E and Var."}]}, "response": "E[X]=1.82; Var[X]=1.565.", "meta": {"id": "WUA_STAT_25K_009291", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.79, s=4.94, n=141, compute CI."}]}, "response": "95% CI ≈ [26.97, 28.61].", "meta": {"id": "WUA_STAT_25K_009292", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.107, sens=0.92, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.408.", "meta": {"id": "WUA_STAT_25K_009293", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.77, s=32.64, n=85, compute CI."}]}, "response": "95% CI ≈ [37.83, 51.71].", "meta": {"id": "WUA_STAT_25K_009294", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.84, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.752.", "meta": {"id": "WUA_STAT_25K_009295", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.84, μ=89.04, σ=2.36, compute z."}]}, "response": "z = -11.102.", "meta": {"id": "WUA_STAT_25K_009296", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.92, s=34.81, n=200, compute CI."}]}, "response": "95% CI ≈ [150.10, 159.74].", "meta": {"id": "WUA_STAT_25K_009297", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.08, μ=81.01, σ=18.41, compute z."}]}, "response": "z = -2.712.", "meta": {"id": "WUA_STAT_25K_009298", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.54, compute E and Var."}]}, "response": "E[X]=7.02; Var[X]=3.229.", "meta": {"id": "WUA_STAT_25K_009299", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.203, sens=0.81, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.579.", "meta": {"id": "WUA_STAT_25K_009300", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.047, sens=0.8, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.158.", "meta": {"id": "WUA_STAT_25K_009301", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.62, μ=84.46, σ=9.93, compute z."}]}, "response": "z = -7.637.", "meta": {"id": "WUA_STAT_25K_009302", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.13, compute E and Var."}]}, "response": "E[X]=0.91; Var[X]=0.792.", "meta": {"id": "WUA_STAT_25K_009303", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.06, compute E and Var."}]}, "response": "E[X]=0.66; Var[X]=0.620.", "meta": {"id": "WUA_STAT_25K_009304", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.42, compute E and Var."}]}, "response": "E[X]=13.02; Var[X]=7.552.", "meta": {"id": "WUA_STAT_25K_009305", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.1, compute E and Var."}]}, "response": "E[X]=3.90; Var[X]=3.510.", "meta": {"id": "WUA_STAT_25K_009306", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.37, μ=93.88, σ=15.7, compute z."}]}, "response": "z = -0.797.", "meta": {"id": "WUA_STAT_25K_009307", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.21, compute E and Var."}]}, "response": "E[X]=4.83; Var[X]=3.816.", "meta": {"id": "WUA_STAT_25K_009308", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.55, compute E and Var."}]}, "response": "E[X]=7.70; Var[X]=3.465.", "meta": {"id": "WUA_STAT_25K_009309", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.8, compute E and Var."}]}, "response": "E[X]=23.20; Var[X]=4.640.", "meta": {"id": "WUA_STAT_25K_009310", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.09, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=2.293.", "meta": {"id": "WUA_STAT_25K_009311", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.81, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.422.", "meta": {"id": "WUA_STAT_25K_009312", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.5, s=36.73, n=157, compute CI."}]}, "response": "95% CI ≈ [178.75, 190.25].", "meta": {"id": "WUA_STAT_25K_009313", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.36, compute E and Var."}]}, "response": "E[X]=10.08; Var[X]=6.451.", "meta": {"id": "WUA_STAT_25K_009314", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.77, s=2.93, n=172, compute CI."}]}, "response": "95% CI ≈ [99.33, 100.21].", "meta": {"id": "WUA_STAT_25K_009315", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.93, s=22.37, n=67, compute CI."}]}, "response": "95% CI ≈ [179.57, 190.29].", "meta": {"id": "WUA_STAT_25K_009316", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.6, compute E and Var."}]}, "response": "E[X]=13.80; Var[X]=5.520.", "meta": {"id": "WUA_STAT_25K_009317", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.81, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.328.", "meta": {"id": "WUA_STAT_25K_009318", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.33, s=25.77, n=10, compute CI."}]}, "response": "95% CI ≈ [165.36, 197.30].", "meta": {"id": "WUA_STAT_25K_009319", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.5, compute E and Var."}]}, "response": "E[X]=14.50; Var[X]=7.250.", "meta": {"id": "WUA_STAT_25K_009320", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.57, μ=35.91, σ=22.63, compute z."}]}, "response": "z = 0.825.", "meta": {"id": "WUA_STAT_25K_009321", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.97, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.441.", "meta": {"id": "WUA_STAT_25K_009322", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.058, sens=0.8, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.176.", "meta": {"id": "WUA_STAT_25K_009323", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=42.34, s=3.09, n=190, compute CI."}]}, "response": "95% CI ≈ [41.90, 42.78].", "meta": {"id": "WUA_STAT_25K_009324", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.14, compute E and Var."}]}, "response": "E[X]=3.22; Var[X]=2.769.", "meta": {"id": "WUA_STAT_25K_009325", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.89, s=13.02, n=103, compute CI."}]}, "response": "95% CI ≈ [19.38, 24.40].", "meta": {"id": "WUA_STAT_25K_009326", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.39, s=11.97, n=115, compute CI."}]}, "response": "95% CI ≈ [179.20, 183.58].", "meta": {"id": "WUA_STAT_25K_009327", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=32.12, s=38.98, n=28, compute CI."}]}, "response": "95% CI ≈ [17.68, 46.56].", "meta": {"id": "WUA_STAT_25K_009328", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.077, sens=0.85, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.503.", "meta": {"id": "WUA_STAT_25K_009329", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.1, compute E and Var."}]}, "response": "E[X]=0.90; Var[X]=0.810.", "meta": {"id": "WUA_STAT_25K_009330", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.93, μ=64.83, σ=4.3, compute z."}]}, "response": "z = -11.605.", "meta": {"id": "WUA_STAT_25K_009331", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.66, compute E and Var."}]}, "response": "E[X]=9.24; Var[X]=3.142.", "meta": {"id": "WUA_STAT_25K_009332", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.29, compute E and Var."}]}, "response": "E[X]=8.12; Var[X]=5.765.", "meta": {"id": "WUA_STAT_25K_009333", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.35, compute E and Var."}]}, "response": "E[X]=4.20; Var[X]=2.730.", "meta": {"id": "WUA_STAT_25K_009334", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.11, μ=14.61, σ=10.55, compute z."}]}, "response": "z = -0.900.", "meta": {"id": "WUA_STAT_25K_009335", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.12, compute E and Var."}]}, "response": "E[X]=3.84; Var[X]=3.379.", "meta": {"id": "WUA_STAT_25K_009336", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.29, μ=50.1, σ=15.16, compute z."}]}, "response": "z = -2.296.", "meta": {"id": "WUA_STAT_25K_009337", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.45, μ=64.44, σ=8.2, compute z."}]}, "response": "z = 1.221.", "meta": {"id": "WUA_STAT_25K_009338", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.084, sens=0.87, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.222.", "meta": {"id": "WUA_STAT_25K_009339", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.21, s=1.44, n=101, compute CI."}]}, "response": "95% CI ≈ [30.93, 31.49].", "meta": {"id": "WUA_STAT_25K_009340", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.79, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.125.", "meta": {"id": "WUA_STAT_25K_009341", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=100.52, s=36.68, n=188, compute CI."}]}, "response": "95% CI ≈ [95.28, 105.76].", "meta": {"id": "WUA_STAT_25K_009342", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.93, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.894.", "meta": {"id": "WUA_STAT_25K_009343", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.06, μ=5.59, σ=13.21, compute z."}]}, "response": "z = 6.546.", "meta": {"id": "WUA_STAT_25K_009344", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.62, μ=75.28, σ=8.47, compute z."}]}, "response": "z = -5.745.", "meta": {"id": "WUA_STAT_25K_009345", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.223, sens=0.86, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.673.", "meta": {"id": "WUA_STAT_25K_009346", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.91, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.756.", "meta": {"id": "WUA_STAT_25K_009347", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.26, μ=42.2, σ=13.68, compute z."}]}, "response": "z = -2.115.", "meta": {"id": "WUA_STAT_25K_009348", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.25, compute E and Var."}]}, "response": "E[X]=2.00; Var[X]=1.500.", "meta": {"id": "WUA_STAT_25K_009349", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.15, s=12.6, n=99, compute CI."}]}, "response": "95% CI ≈ [187.67, 192.63].", "meta": {"id": "WUA_STAT_25K_009350", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.39, compute E and Var."}]}, "response": "E[X]=12.48; Var[X]=7.613.", "meta": {"id": "WUA_STAT_25K_009351", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.79, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.372.", "meta": {"id": "WUA_STAT_25K_009352", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.223, sens=0.97, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.582.", "meta": {"id": "WUA_STAT_25K_009353", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.201, sens=0.92, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.577.", "meta": {"id": "WUA_STAT_25K_009354", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.085, sens=0.92, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.228.", "meta": {"id": "WUA_STAT_25K_009355", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.12, s=11.07, n=64, compute CI."}]}, "response": "95% CI ≈ [103.41, 108.83].", "meta": {"id": "WUA_STAT_25K_009356", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.15, μ=37.81, σ=21.48, compute z."}]}, "response": "z = 1.273.", "meta": {"id": "WUA_STAT_25K_009357", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.54, μ=65.12, σ=24.91, compute z."}]}, "response": "z = -2.432.", "meta": {"id": "WUA_STAT_25K_009358", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.031, sens=0.75, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.094.", "meta": {"id": "WUA_STAT_25K_009359", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.78, μ=20.59, σ=19.76, compute z."}]}, "response": "z = 3.147.", "meta": {"id": "WUA_STAT_25K_009360", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.73, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.458.", "meta": {"id": "WUA_STAT_25K_009361", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.47, compute E and Var."}]}, "response": "E[X]=5.64; Var[X]=2.989.", "meta": {"id": "WUA_STAT_25K_009362", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.83, s=1.74, n=168, compute CI."}]}, "response": "95% CI ≈ [61.57, 62.09].", "meta": {"id": "WUA_STAT_25K_009363", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.16, s=30.7, n=158, compute CI."}]}, "response": "95% CI ≈ [57.37, 66.95].", "meta": {"id": "WUA_STAT_25K_009364", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.28, s=13.23, n=176, compute CI."}]}, "response": "95% CI ≈ [11.33, 15.23].", "meta": {"id": "WUA_STAT_25K_009365", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.246, sens=0.8, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.482.", "meta": {"id": "WUA_STAT_25K_009366", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.18, s=3.92, n=131, compute CI."}]}, "response": "95% CI ≈ [16.51, 17.85].", "meta": {"id": "WUA_STAT_25K_009367", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.054, sens=0.79, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.170.", "meta": {"id": "WUA_STAT_25K_009368", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.55, μ=19.83, σ=5.7, compute z."}]}, "response": "z = 11.004.", "meta": {"id": "WUA_STAT_25K_009369", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.91, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.084.", "meta": {"id": "WUA_STAT_25K_009370", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.026, sens=0.73, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.093.", "meta": {"id": "WUA_STAT_25K_009371", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.39, μ=81.91, σ=22.53, compute z."}]}, "response": "z = -3.485.", "meta": {"id": "WUA_STAT_25K_009372", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.73, compute E and Var."}]}, "response": "E[X]=6.57; Var[X]=1.774.", "meta": {"id": "WUA_STAT_25K_009373", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.75, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.207.", "meta": {"id": "WUA_STAT_25K_009374", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.96, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.342.", "meta": {"id": "WUA_STAT_25K_009375", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.47, μ=93.73, σ=8.28, compute z."}]}, "response": "z = -0.394.", "meta": {"id": "WUA_STAT_25K_009376", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.018, sens=0.93, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.096.", "meta": {"id": "WUA_STAT_25K_009377", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.36, s=28.68, n=104, compute CI."}]}, "response": "95% CI ≈ [157.85, 168.87].", "meta": {"id": "WUA_STAT_25K_009378", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.9, s=30.75, n=123, compute CI."}]}, "response": "95% CI ≈ [135.47, 146.33].", "meta": {"id": "WUA_STAT_25K_009379", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.58, compute E and Var."}]}, "response": "E[X]=18.56; Var[X]=7.795.", "meta": {"id": "WUA_STAT_25K_009380", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.72, s=27.24, n=34, compute CI."}]}, "response": "95% CI ≈ [60.56, 78.88].", "meta": {"id": "WUA_STAT_25K_009381", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.03, μ=6.8, σ=23.12, compute z."}]}, "response": "z = 1.654.", "meta": {"id": "WUA_STAT_25K_009382", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.61, s=18.97, n=170, compute CI."}]}, "response": "95% CI ≈ [54.76, 60.46].", "meta": {"id": "WUA_STAT_25K_009383", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.38, μ=29.52, σ=14.19, compute z."}]}, "response": "z = 2.175.", "meta": {"id": "WUA_STAT_25K_009384", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.65, μ=73.75, σ=12.77, compute z."}]}, "response": "z = -1.182.", "meta": {"id": "WUA_STAT_25K_009385", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.5, s=27.55, n=130, compute CI."}]}, "response": "95% CI ≈ [74.76, 84.24].", "meta": {"id": "WUA_STAT_25K_009386", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.81, s=10.67, n=178, compute CI."}]}, "response": "95% CI ≈ [192.24, 195.38].", "meta": {"id": "WUA_STAT_25K_009387", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.24, compute E and Var."}]}, "response": "E[X]=3.84; Var[X]=2.918.", "meta": {"id": "WUA_STAT_25K_009388", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=171.72, s=14.76, n=179, compute CI."}]}, "response": "95% CI ≈ [169.56, 173.88].", "meta": {"id": "WUA_STAT_25K_009389", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.52, s=7.37, n=113, compute CI."}]}, "response": "95% CI ≈ [152.16, 154.88].", "meta": {"id": "WUA_STAT_25K_009390", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.09, μ=99.58, σ=8.43, compute z."}]}, "response": "z = -3.735.", "meta": {"id": "WUA_STAT_25K_009391", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.79, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.788.", "meta": {"id": "WUA_STAT_25K_009392", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.092, sens=0.83, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.244.", "meta": {"id": "WUA_STAT_25K_009393", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.63, compute E and Var."}]}, "response": "E[X]=18.27; Var[X]=6.760.", "meta": {"id": "WUA_STAT_25K_009394", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.95, s=8.9, n=170, compute CI."}]}, "response": "95% CI ≈ [171.61, 174.29].", "meta": {"id": "WUA_STAT_25K_009395", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.16, compute E and Var."}]}, "response": "E[X]=3.52; Var[X]=2.957.", "meta": {"id": "WUA_STAT_25K_009396", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.54, compute E and Var."}]}, "response": "E[X]=18.90; Var[X]=8.694.", "meta": {"id": "WUA_STAT_25K_009397", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.63, compute E and Var."}]}, "response": "E[X]=23.94; Var[X]=8.858.", "meta": {"id": "WUA_STAT_25K_009398", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=124.58, s=16.05, n=61, compute CI."}]}, "response": "95% CI ≈ [120.55, 128.61].", "meta": {"id": "WUA_STAT_25K_009399", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.76, compute E and Var."}]}, "response": "E[X]=14.44; Var[X]=3.466.", "meta": {"id": "WUA_STAT_25K_009400", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.77, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.438.", "meta": {"id": "WUA_STAT_25K_009401", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.35, compute E and Var."}]}, "response": "E[X]=7.00; Var[X]=4.550.", "meta": {"id": "WUA_STAT_25K_009402", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.048, sens=0.77, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.126.", "meta": {"id": "WUA_STAT_25K_009403", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.55, compute E and Var."}]}, "response": "E[X]=14.85; Var[X]=6.683.", "meta": {"id": "WUA_STAT_25K_009404", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.79, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.461.", "meta": {"id": "WUA_STAT_25K_009405", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.42, μ=88.06, σ=15.3, compute z."}]}, "response": "z = 0.024.", "meta": {"id": "WUA_STAT_25K_009406", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.22, s=20.84, n=12, compute CI."}]}, "response": "95% CI ≈ [145.43, 169.01].", "meta": {"id": "WUA_STAT_25K_009407", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.89, μ=8.66, σ=17.84, compute z."}]}, "response": "z = 0.405.", "meta": {"id": "WUA_STAT_25K_009408", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.094, sens=0.75, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.609.", "meta": {"id": "WUA_STAT_25K_009409", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.14, s=23.98, n=126, compute CI."}]}, "response": "95% CI ≈ [181.95, 190.33].", "meta": {"id": "WUA_STAT_25K_009410", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.74, s=20.84, n=118, compute CI."}]}, "response": "95% CI ≈ [59.98, 67.50].", "meta": {"id": "WUA_STAT_25K_009411", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.68, compute E and Var."}]}, "response": "E[X]=14.96; Var[X]=4.787.", "meta": {"id": "WUA_STAT_25K_009412", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.26, compute E and Var."}]}, "response": "E[X]=3.38; Var[X]=2.501.", "meta": {"id": "WUA_STAT_25K_009413", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.8, μ=55.45, σ=15.77, compute z."}]}, "response": "z = 0.529.", "meta": {"id": "WUA_STAT_25K_009414", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.19, s=27.98, n=58, compute CI."}]}, "response": "95% CI ≈ [178.99, 193.39].", "meta": {"id": "WUA_STAT_25K_009415", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.71, s=24.98, n=170, compute CI."}]}, "response": "95% CI ≈ [37.95, 45.47].", "meta": {"id": "WUA_STAT_25K_009416", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.93, μ=18.43, σ=15.15, compute z."}]}, "response": "z = 3.135.", "meta": {"id": "WUA_STAT_25K_009417", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.195, sens=0.79, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.732.", "meta": {"id": "WUA_STAT_25K_009418", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.42, μ=90.43, σ=17.82, compute z."}]}, "response": "z = 0.168.", "meta": {"id": "WUA_STAT_25K_009419", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.42, compute E and Var."}]}, "response": "E[X]=9.24; Var[X]=5.359.", "meta": {"id": "WUA_STAT_25K_009420", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.53, compute E and Var."}]}, "response": "E[X]=6.89; Var[X]=3.238.", "meta": {"id": "WUA_STAT_25K_009421", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.13, compute E and Var."}]}, "response": "E[X]=4.16; Var[X]=3.619.", "meta": {"id": "WUA_STAT_25K_009422", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.047, sens=0.77, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.240.", "meta": {"id": "WUA_STAT_25K_009423", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.32, μ=22.46, σ=20.56, compute z."}]}, "response": "z = 2.863.", "meta": {"id": "WUA_STAT_25K_009424", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.84, s=8.21, n=196, compute CI."}]}, "response": "95% CI ≈ [121.69, 123.99].", "meta": {"id": "WUA_STAT_25K_009425", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.74, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.659.", "meta": {"id": "WUA_STAT_25K_009426", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.52, compute E and Var."}]}, "response": "E[X]=3.64; Var[X]=1.747.", "meta": {"id": "WUA_STAT_25K_009427", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.79, s=34.09, n=29, compute CI."}]}, "response": "95% CI ≈ [128.38, 153.20].", "meta": {"id": "WUA_STAT_25K_009428", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.68, μ=59.48, σ=21.0, compute z."}]}, "response": "z = -1.467.", "meta": {"id": "WUA_STAT_25K_009429", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.05, sens=0.75, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.128.", "meta": {"id": "WUA_STAT_25K_009430", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.059, sens=0.74, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.367.", "meta": {"id": "WUA_STAT_25K_009431", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.065, sens=0.91, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.260.", "meta": {"id": "WUA_STAT_25K_009432", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.98, μ=8.79, σ=21.24, compute z."}]}, "response": "z = 1.845.", "meta": {"id": "WUA_STAT_25K_009433", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.08, compute E and Var."}]}, "response": "E[X]=1.12; Var[X]=1.030.", "meta": {"id": "WUA_STAT_25K_009434", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.34, μ=32.89, σ=1.39, compute z."}]}, "response": "z = 2.482.", "meta": {"id": "WUA_STAT_25K_009435", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.22, μ=85.51, σ=9.12, compute z."}]}, "response": "z = -8.804.", "meta": {"id": "WUA_STAT_25K_009436", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.59, s=14.49, n=63, compute CI."}]}, "response": "95% CI ≈ [85.01, 92.17].", "meta": {"id": "WUA_STAT_25K_009437", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.88, s=9.96, n=171, compute CI."}]}, "response": "95% CI ≈ [10.39, 13.37].", "meta": {"id": "WUA_STAT_25K_009438", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.52, μ=84.47, σ=23.45, compute z."}]}, "response": "z = -2.770.", "meta": {"id": "WUA_STAT_25K_009439", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.78, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.394.", "meta": {"id": "WUA_STAT_25K_009440", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.31, s=20.46, n=156, compute CI."}]}, "response": "95% CI ≈ [114.10, 120.52].", "meta": {"id": "WUA_STAT_25K_009441", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.8, μ=42.22, σ=10.98, compute z."}]}, "response": "z = -3.317.", "meta": {"id": "WUA_STAT_25K_009442", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.128, sens=0.77, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.790.", "meta": {"id": "WUA_STAT_25K_009443", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.49, s=16.86, n=88, compute CI."}]}, "response": "95% CI ≈ [108.97, 116.01].", "meta": {"id": "WUA_STAT_25K_009444", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.92, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.837.", "meta": {"id": "WUA_STAT_25K_009445", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.75, compute E and Var."}]}, "response": "E[X]=17.25; Var[X]=4.312.", "meta": {"id": "WUA_STAT_25K_009446", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.06, compute E and Var."}]}, "response": "E[X]=1.50; Var[X]=1.410.", "meta": {"id": "WUA_STAT_25K_009447", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.62, s=14.28, n=61, compute CI."}]}, "response": "95% CI ≈ [111.04, 118.20].", "meta": {"id": "WUA_STAT_25K_009448", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.6, s=11.44, n=181, compute CI."}]}, "response": "95% CI ≈ [16.93, 20.27].", "meta": {"id": "WUA_STAT_25K_009449", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.53, compute E and Var."}]}, "response": "E[X]=6.89; Var[X]=3.238.", "meta": {"id": "WUA_STAT_25K_009450", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.09, μ=75.91, σ=4.01, compute z."}]}, "response": "z = -16.663.", "meta": {"id": "WUA_STAT_25K_009451", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.06, μ=67.82, σ=3.42, compute z."}]}, "response": "z = -19.813.", "meta": {"id": "WUA_STAT_25K_009452", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.22, compute E and Var."}]}, "response": "E[X]=5.28; Var[X]=4.118.", "meta": {"id": "WUA_STAT_25K_009453", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.77, compute E and Var."}]}, "response": "E[X]=21.56; Var[X]=4.959.", "meta": {"id": "WUA_STAT_25K_009454", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.08, μ=17.06, σ=17.31, compute z."}]}, "response": "z = 0.463.", "meta": {"id": "WUA_STAT_25K_009455", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.134, sens=0.8, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.674.", "meta": {"id": "WUA_STAT_25K_009456", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.54, s=19.75, n=87, compute CI."}]}, "response": "95% CI ≈ [119.39, 127.69].", "meta": {"id": "WUA_STAT_25K_009457", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.17, compute E and Var."}]}, "response": "E[X]=2.04; Var[X]=1.693.", "meta": {"id": "WUA_STAT_25K_009458", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.208, sens=0.95, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.657.", "meta": {"id": "WUA_STAT_25K_009459", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.95, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.330.", "meta": {"id": "WUA_STAT_25K_009460", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.23, s=14.43, n=147, compute CI."}]}, "response": "95% CI ≈ [191.90, 196.56].", "meta": {"id": "WUA_STAT_25K_009461", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.029, sens=0.76, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.090.", "meta": {"id": "WUA_STAT_25K_009462", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.23, compute E and Var."}]}, "response": "E[X]=6.90; Var[X]=5.313.", "meta": {"id": "WUA_STAT_25K_009463", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.218, sens=0.85, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.467.", "meta": {"id": "WUA_STAT_25K_009464", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.96, μ=86.67, σ=3.32, compute z."}]}, "response": "z = -25.214.", "meta": {"id": "WUA_STAT_25K_009465", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.12, compute E and Var."}]}, "response": "E[X]=1.56; Var[X]=1.373.", "meta": {"id": "WUA_STAT_25K_009466", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.22, compute E and Var."}]}, "response": "E[X]=1.76; Var[X]=1.373.", "meta": {"id": "WUA_STAT_25K_009467", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.221, sens=0.75, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.423.", "meta": {"id": "WUA_STAT_25K_009468", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.32, μ=75.67, σ=3.65, compute z."}]}, "response": "z = -6.945.", "meta": {"id": "WUA_STAT_25K_009469", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.1, compute E and Var."}]}, "response": "E[X]=1.50; Var[X]=1.350.", "meta": {"id": "WUA_STAT_25K_009470", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=95.57, s=10.43, n=28, compute CI."}]}, "response": "95% CI ≈ [91.71, 99.43].", "meta": {"id": "WUA_STAT_25K_009471", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.23, compute E and Var."}]}, "response": "E[X]=2.76; Var[X]=2.125.", "meta": {"id": "WUA_STAT_25K_009472", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.02, s=3.51, n=18, compute CI."}]}, "response": "95% CI ≈ [33.40, 36.64].", "meta": {"id": "WUA_STAT_25K_009473", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.2, μ=15.26, σ=3.68, compute z."}]}, "response": "z = 4.875.", "meta": {"id": "WUA_STAT_25K_009474", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.124, sens=0.71, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.589.", "meta": {"id": "WUA_STAT_25K_009475", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.35, s=18.19, n=162, compute CI."}]}, "response": "95% CI ≈ [28.55, 34.15].", "meta": {"id": "WUA_STAT_25K_009476", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=10.78, s=4.6, n=152, compute CI."}]}, "response": "95% CI ≈ [10.05, 11.51].", "meta": {"id": "WUA_STAT_25K_009477", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.78, compute E and Var."}]}, "response": "E[X]=26.52; Var[X]=5.834.", "meta": {"id": "WUA_STAT_25K_009478", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.4, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_009479", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.45, μ=61.81, σ=15.79, compute z."}]}, "response": "z = 1.054.", "meta": {"id": "WUA_STAT_25K_009480", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.29, μ=13.87, σ=16.82, compute z."}]}, "response": "z = 2.522.", "meta": {"id": "WUA_STAT_25K_009481", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.67, compute E and Var."}]}, "response": "E[X]=6.03; Var[X]=1.990.", "meta": {"id": "WUA_STAT_25K_009482", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.7, s=4.25, n=180, compute CI."}]}, "response": "95% CI ≈ [90.08, 91.32].", "meta": {"id": "WUA_STAT_25K_009483", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.09, s=24.62, n=33, compute CI."}]}, "response": "95% CI ≈ [121.69, 138.49].", "meta": {"id": "WUA_STAT_25K_009484", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.35, s=22.35, n=25, compute CI."}]}, "response": "95% CI ≈ [184.59, 202.11].", "meta": {"id": "WUA_STAT_25K_009485", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.71, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.401.", "meta": {"id": "WUA_STAT_25K_009486", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.27, compute E and Var."}]}, "response": "E[X]=4.59; Var[X]=3.351.", "meta": {"id": "WUA_STAT_25K_009487", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.072, sens=0.83, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.392.", "meta": {"id": "WUA_STAT_25K_009488", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.75, compute E and Var."}]}, "response": "E[X]=9.00; Var[X]=2.250.", "meta": {"id": "WUA_STAT_25K_009489", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.038, sens=0.97, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.117.", "meta": {"id": "WUA_STAT_25K_009490", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.208, sens=0.92, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.535.", "meta": {"id": "WUA_STAT_25K_009491", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.3, s=15.33, n=56, compute CI."}]}, "response": "95% CI ≈ [41.28, 49.32].", "meta": {"id": "WUA_STAT_25K_009492", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.48, μ=86.25, σ=14.9, compute z."}]}, "response": "z = -0.253.", "meta": {"id": "WUA_STAT_25K_009493", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=26.7, s=9.49, n=92, compute CI."}]}, "response": "95% CI ≈ [24.76, 28.64].", "meta": {"id": "WUA_STAT_25K_009494", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.71, compute E and Var."}]}, "response": "E[X]=3.55; Var[X]=1.030.", "meta": {"id": "WUA_STAT_25K_009495", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.28, s=39.91, n=166, compute CI."}]}, "response": "95% CI ≈ [61.21, 73.35].", "meta": {"id": "WUA_STAT_25K_009496", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.048, sens=0.83, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.126.", "meta": {"id": "WUA_STAT_25K_009497", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.45, compute E and Var."}]}, "response": "E[X]=8.55; Var[X]=4.703.", "meta": {"id": "WUA_STAT_25K_009498", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.37, s=28.37, n=128, compute CI."}]}, "response": "95% CI ≈ [130.46, 140.28].", "meta": {"id": "WUA_STAT_25K_009499", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.85, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.730.", "meta": {"id": "WUA_STAT_25K_009500", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.58, s=29.88, n=80, compute CI."}]}, "response": "95% CI ≈ [132.03, 145.13].", "meta": {"id": "WUA_STAT_25K_009501", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.87, μ=56.51, σ=12.73, compute z."}]}, "response": "z = 1.128.", "meta": {"id": "WUA_STAT_25K_009502", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.84, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.838.", "meta": {"id": "WUA_STAT_25K_009503", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.45, μ=50.47, σ=6.15, compute z."}]}, "response": "z = -3.255.", "meta": {"id": "WUA_STAT_25K_009504", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.43, μ=24.42, σ=12.75, compute z."}]}, "response": "z = -1.019.", "meta": {"id": "WUA_STAT_25K_009505", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.97, s=26.5, n=150, compute CI."}]}, "response": "95% CI ≈ [149.73, 158.21].", "meta": {"id": "WUA_STAT_25K_009506", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.48, compute E and Var."}]}, "response": "E[X]=12.96; Var[X]=6.739.", "meta": {"id": "WUA_STAT_25K_009507", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.25, μ=63.59, σ=21.26, compute z."}]}, "response": "z = -0.157.", "meta": {"id": "WUA_STAT_25K_009508", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.34, s=19.62, n=116, compute CI."}]}, "response": "95% CI ≈ [45.77, 52.91].", "meta": {"id": "WUA_STAT_25K_009509", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.34, μ=18.65, σ=19.34, compute z."}]}, "response": "z = -0.430.", "meta": {"id": "WUA_STAT_25K_009510", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.22, μ=15.23, σ=14.29, compute z."}]}, "response": "z = 1.399.", "meta": {"id": "WUA_STAT_25K_009511", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.31, μ=2.82, σ=19.66, compute z."}]}, "response": "z = 3.738.", "meta": {"id": "WUA_STAT_25K_009512", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.14, compute E and Var."}]}, "response": "E[X]=2.80; Var[X]=2.408.", "meta": {"id": "WUA_STAT_25K_009513", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.75, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.309.", "meta": {"id": "WUA_STAT_25K_009514", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.49, μ=10.8, σ=22.18, compute z."}]}, "response": "z = 3.638.", "meta": {"id": "WUA_STAT_25K_009515", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.09, μ=71.94, σ=10.33, compute z."}]}, "response": "z = -5.407.", "meta": {"id": "WUA_STAT_25K_009516", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.33, compute E and Var."}]}, "response": "E[X]=11.22; Var[X]=7.517.", "meta": {"id": "WUA_STAT_25K_009517", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.93, μ=17.59, σ=5.78, compute z."}]}, "response": "z = 12.516.", "meta": {"id": "WUA_STAT_25K_009518", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.72, compute E and Var."}]}, "response": "E[X]=25.20; Var[X]=7.056.", "meta": {"id": "WUA_STAT_25K_009519", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.73, compute E and Var."}]}, "response": "E[X]=28.47; Var[X]=7.687.", "meta": {"id": "WUA_STAT_25K_009520", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.33, s=19.94, n=19, compute CI."}]}, "response": "95% CI ≈ [71.36, 89.30].", "meta": {"id": "WUA_STAT_25K_009521", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.47, compute E and Var."}]}, "response": "E[X]=4.23; Var[X]=2.242.", "meta": {"id": "WUA_STAT_25K_009522", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.87, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.407.", "meta": {"id": "WUA_STAT_25K_009523", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.091, sens=0.77, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.251.", "meta": {"id": "WUA_STAT_25K_009524", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.89, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.753.", "meta": {"id": "WUA_STAT_25K_009525", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.072, sens=0.89, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.301.", "meta": {"id": "WUA_STAT_25K_009526", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.61, compute E and Var."}]}, "response": "E[X]=4.27; Var[X]=1.665.", "meta": {"id": "WUA_STAT_25K_009527", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.39, s=13.75, n=54, compute CI."}]}, "response": "95% CI ≈ [183.72, 191.06].", "meta": {"id": "WUA_STAT_25K_009528", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.104, sens=0.8, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.481.", "meta": {"id": "WUA_STAT_25K_009529", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.07, compute E and Var."}]}, "response": "E[X]=1.89; Var[X]=1.758.", "meta": {"id": "WUA_STAT_25K_009530", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.76, s=23.12, n=141, compute CI."}]}, "response": "95% CI ≈ [119.94, 127.58].", "meta": {"id": "WUA_STAT_25K_009531", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.23, s=4.65, n=184, compute CI."}]}, "response": "95% CI ≈ [76.56, 77.90].", "meta": {"id": "WUA_STAT_25K_009532", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.85, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.432.", "meta": {"id": "WUA_STAT_25K_009533", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.13, compute E and Var."}]}, "response": "E[X]=4.29; Var[X]=3.732.", "meta": {"id": "WUA_STAT_25K_009534", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.23, sens=0.82, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.505.", "meta": {"id": "WUA_STAT_25K_009535", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.45, compute E and Var."}]}, "response": "E[X]=7.65; Var[X]=4.208.", "meta": {"id": "WUA_STAT_25K_009536", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.08, μ=9.04, σ=18.87, compute z."}]}, "response": "z = 3.288.", "meta": {"id": "WUA_STAT_25K_009537", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.3, s=24.59, n=67, compute CI."}]}, "response": "95% CI ≈ [139.41, 151.19].", "meta": {"id": "WUA_STAT_25K_009538", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.37, μ=7.22, σ=9.99, compute z."}]}, "response": "z = 0.616.", "meta": {"id": "WUA_STAT_25K_009539", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.4, s=16.57, n=157, compute CI."}]}, "response": "95% CI ≈ [119.81, 124.99].", "meta": {"id": "WUA_STAT_25K_009540", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.062, sens=0.73, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.376.", "meta": {"id": "WUA_STAT_25K_009541", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.4, s=7.63, n=138, compute CI."}]}, "response": "95% CI ≈ [165.13, 167.67].", "meta": {"id": "WUA_STAT_25K_009542", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.71, s=20.31, n=71, compute CI."}]}, "response": "95% CI ≈ [20.99, 30.43].", "meta": {"id": "WUA_STAT_25K_009543", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.55, μ=29.58, σ=10.75, compute z."}]}, "response": "z = 0.648.", "meta": {"id": "WUA_STAT_25K_009544", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.25, compute E and Var."}]}, "response": "E[X]=9.00; Var[X]=6.750.", "meta": {"id": "WUA_STAT_25K_009545", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.187, sens=0.72, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.372.", "meta": {"id": "WUA_STAT_25K_009546", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.46, μ=2.01, σ=24.32, compute z."}]}, "response": "z = -0.023.", "meta": {"id": "WUA_STAT_25K_009547", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.033, sens=0.88, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.143.", "meta": {"id": "WUA_STAT_25K_009548", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.98, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.473.", "meta": {"id": "WUA_STAT_25K_009549", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.17, compute E and Var."}]}, "response": "E[X]=5.44; Var[X]=4.515.", "meta": {"id": "WUA_STAT_25K_009550", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.15, s=11.89, n=131, compute CI."}]}, "response": "95% CI ≈ [118.11, 122.19].", "meta": {"id": "WUA_STAT_25K_009551", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.85, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.704.", "meta": {"id": "WUA_STAT_25K_009552", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.48, s=24.37, n=47, compute CI."}]}, "response": "95% CI ≈ [14.51, 28.45].", "meta": {"id": "WUA_STAT_25K_009553", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.75, compute E and Var."}]}, "response": "E[X]=24.00; Var[X]=6.000.", "meta": {"id": "WUA_STAT_25K_009554", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.043, sens=0.86, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.121.", "meta": {"id": "WUA_STAT_25K_009555", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.08, s=22.24, n=109, compute CI."}]}, "response": "95% CI ≈ [17.90, 26.26].", "meta": {"id": "WUA_STAT_25K_009556", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.63, s=1.19, n=200, compute CI."}]}, "response": "95% CI ≈ [153.47, 153.79].", "meta": {"id": "WUA_STAT_25K_009557", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.37, μ=9.62, σ=11.99, compute z."}]}, "response": "z = 3.399.", "meta": {"id": "WUA_STAT_25K_009558", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.32, μ=66.29, σ=6.7, compute z."}]}, "response": "z = 1.348.", "meta": {"id": "WUA_STAT_25K_009559", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.17, s=34.06, n=40, compute CI."}]}, "response": "95% CI ≈ [33.61, 54.73].", "meta": {"id": "WUA_STAT_25K_009560", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.7, μ=30.27, σ=16.93, compute z."}]}, "response": "z = -0.742.", "meta": {"id": "WUA_STAT_25K_009561", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.75, compute E and Var."}]}, "response": "E[X]=9.75; Var[X]=2.438.", "meta": {"id": "WUA_STAT_25K_009562", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=56.91, s=24.92, n=90, compute CI."}]}, "response": "95% CI ≈ [51.76, 62.06].", "meta": {"id": "WUA_STAT_25K_009563", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.1, compute E and Var."}]}, "response": "E[X]=2.80; Var[X]=2.520.", "meta": {"id": "WUA_STAT_25K_009564", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.76, compute E and Var."}]}, "response": "E[X]=27.36; Var[X]=6.566.", "meta": {"id": "WUA_STAT_25K_009565", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.93, μ=40.59, σ=17.5, compute z."}]}, "response": "z = 2.477.", "meta": {"id": "WUA_STAT_25K_009566", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.91, μ=87.7, σ=2.05, compute z."}]}, "response": "z = -40.385.", "meta": {"id": "WUA_STAT_25K_009567", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.29, s=38.87, n=162, compute CI."}]}, "response": "95% CI ≈ [173.30, 185.28].", "meta": {"id": "WUA_STAT_25K_009568", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.64, compute E and Var."}]}, "response": "E[X]=7.68; Var[X]=2.765.", "meta": {"id": "WUA_STAT_25K_009569", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.7, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.285.", "meta": {"id": "WUA_STAT_25K_009570", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.037, sens=0.78, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.187.", "meta": {"id": "WUA_STAT_25K_009571", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.56, s=39.55, n=120, compute CI."}]}, "response": "95% CI ≈ [112.48, 126.64].", "meta": {"id": "WUA_STAT_25K_009572", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.91, s=22.36, n=130, compute CI."}]}, "response": "95% CI ≈ [120.07, 127.75].", "meta": {"id": "WUA_STAT_25K_009573", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.56, compute E and Var."}]}, "response": "E[X]=7.84; Var[X]=3.450.", "meta": {"id": "WUA_STAT_25K_009574", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.31, μ=82.49, σ=11.42, compute z."}]}, "response": "z = -1.242.", "meta": {"id": "WUA_STAT_25K_009575", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.05, compute E and Var."}]}, "response": "E[X]=0.80; Var[X]=0.760.", "meta": {"id": "WUA_STAT_25K_009576", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.15, μ=80.0, σ=24.4, compute z."}]}, "response": "z = -1.797.", "meta": {"id": "WUA_STAT_25K_009577", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.097, sens=0.91, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.328.", "meta": {"id": "WUA_STAT_25K_009578", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=16.43, s=21.57, n=180, compute CI."}]}, "response": "95% CI ≈ [13.28, 19.58].", "meta": {"id": "WUA_STAT_25K_009579", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.53, compute E and Var."}]}, "response": "E[X]=6.89; Var[X]=3.238.", "meta": {"id": "WUA_STAT_25K_009580", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.96, s=35.59, n=104, compute CI."}]}, "response": "95% CI ≈ [139.12, 152.80].", "meta": {"id": "WUA_STAT_25K_009581", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.35, compute E and Var."}]}, "response": "E[X]=12.25; Var[X]=7.963.", "meta": {"id": "WUA_STAT_25K_009582", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.46, compute E and Var."}]}, "response": "E[X]=10.12; Var[X]=5.465.", "meta": {"id": "WUA_STAT_25K_009583", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.82, μ=74.53, σ=15.97, compute z."}]}, "response": "z = -3.363.", "meta": {"id": "WUA_STAT_25K_009584", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.13, compute E and Var."}]}, "response": "E[X]=2.60; Var[X]=2.262.", "meta": {"id": "WUA_STAT_25K_009585", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.73, compute E and Var."}]}, "response": "E[X]=17.52; Var[X]=4.730.", "meta": {"id": "WUA_STAT_25K_009586", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.62, compute E and Var."}]}, "response": "E[X]=11.16; Var[X]=4.241.", "meta": {"id": "WUA_STAT_25K_009587", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.62, s=26.43, n=81, compute CI."}]}, "response": "95% CI ≈ [116.86, 128.38].", "meta": {"id": "WUA_STAT_25K_009588", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.24, s=28.66, n=54, compute CI."}]}, "response": "95% CI ≈ [153.60, 168.88].", "meta": {"id": "WUA_STAT_25K_009589", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.71, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.207.", "meta": {"id": "WUA_STAT_25K_009590", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.06, μ=62.39, σ=16.57, compute z."}]}, "response": "z = -0.563.", "meta": {"id": "WUA_STAT_25K_009591", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.057, sens=0.91, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.579.", "meta": {"id": "WUA_STAT_25K_009592", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.15, s=37.19, n=69, compute CI."}]}, "response": "95% CI ≈ [41.37, 58.93].", "meta": {"id": "WUA_STAT_25K_009593", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.25, compute E and Var."}]}, "response": "E[X]=9.25; Var[X]=6.938.", "meta": {"id": "WUA_STAT_25K_009594", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.111, sens=0.91, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.558.", "meta": {"id": "WUA_STAT_25K_009595", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.76, compute E and Var."}]}, "response": "E[X]=15.20; Var[X]=3.648.", "meta": {"id": "WUA_STAT_25K_009596", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.79, compute E and Var."}]}, "response": "E[X]=16.59; Var[X]=3.484.", "meta": {"id": "WUA_STAT_25K_009597", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.4, μ=13.78, σ=12.67, compute z."}]}, "response": "z = 0.286.", "meta": {"id": "WUA_STAT_25K_009598", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.179, sens=0.91, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.908.", "meta": {"id": "WUA_STAT_25K_009599", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.167, sens=0.99, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.539.", "meta": {"id": "WUA_STAT_25K_009600", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.89, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.295.", "meta": {"id": "WUA_STAT_25K_009601", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.1, compute E and Var."}]}, "response": "E[X]=3.40; Var[X]=3.060.", "meta": {"id": "WUA_STAT_25K_009602", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.31, μ=75.1, σ=22.75, compute z."}]}, "response": "z = -2.804.", "meta": {"id": "WUA_STAT_25K_009603", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.86, μ=81.29, σ=10.67, compute z."}]}, "response": "z = 1.647.", "meta": {"id": "WUA_STAT_25K_009604", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.83, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.774.", "meta": {"id": "WUA_STAT_25K_009605", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.92, μ=18.5, σ=24.32, compute z."}]}, "response": "z = 0.922.", "meta": {"id": "WUA_STAT_25K_009606", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.84, μ=86.64, σ=5.09, compute z."}]}, "response": "z = -3.694.", "meta": {"id": "WUA_STAT_25K_009607", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.59, compute E and Var."}]}, "response": "E[X]=23.60; Var[X]=9.676.", "meta": {"id": "WUA_STAT_25K_009608", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.43, s=35.0, n=70, compute CI."}]}, "response": "95% CI ≈ [102.23, 118.63].", "meta": {"id": "WUA_STAT_25K_009609", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.78, s=38.34, n=135, compute CI."}]}, "response": "95% CI ≈ [139.31, 152.25].", "meta": {"id": "WUA_STAT_25K_009610", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.016, sens=0.73, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.078.", "meta": {"id": "WUA_STAT_25K_009611", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.43, s=37.92, n=66, compute CI."}]}, "response": "95% CI ≈ [26.28, 44.58].", "meta": {"id": "WUA_STAT_25K_009612", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.15, s=15.39, n=77, compute CI."}]}, "response": "95% CI ≈ [169.71, 176.59].", "meta": {"id": "WUA_STAT_25K_009613", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.76, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.481.", "meta": {"id": "WUA_STAT_25K_009614", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.9, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.884.", "meta": {"id": "WUA_STAT_25K_009615", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.98, s=18.46, n=102, compute CI."}]}, "response": "95% CI ≈ [183.40, 190.56].", "meta": {"id": "WUA_STAT_25K_009616", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.36, compute E and Var."}]}, "response": "E[X]=9.00; Var[X]=5.760.", "meta": {"id": "WUA_STAT_25K_009617", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.74, compute E and Var."}]}, "response": "E[X]=17.02; Var[X]=4.425.", "meta": {"id": "WUA_STAT_25K_009618", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.78, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.603.", "meta": {"id": "WUA_STAT_25K_009619", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.6, s=15.99, n=69, compute CI."}]}, "response": "95% CI ≈ [74.83, 82.37].", "meta": {"id": "WUA_STAT_25K_009620", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.29, s=24.5, n=148, compute CI."}]}, "response": "95% CI ≈ [33.34, 41.24].", "meta": {"id": "WUA_STAT_25K_009621", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.212, sens=0.8, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.956.", "meta": {"id": "WUA_STAT_25K_009622", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=141.06, s=19.33, n=109, compute CI."}]}, "response": "95% CI ≈ [137.43, 144.69].", "meta": {"id": "WUA_STAT_25K_009623", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.41, s=7.94, n=196, compute CI."}]}, "response": "95% CI ≈ [173.30, 175.52].", "meta": {"id": "WUA_STAT_25K_009624", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.51, compute E and Var."}]}, "response": "E[X]=11.73; Var[X]=5.748.", "meta": {"id": "WUA_STAT_25K_009625", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.49, s=34.94, n=196, compute CI."}]}, "response": "95% CI ≈ [50.60, 60.38].", "meta": {"id": "WUA_STAT_25K_009626", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.93, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.548.", "meta": {"id": "WUA_STAT_25K_009627", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.73, s=18.2, n=138, compute CI."}]}, "response": "95% CI ≈ [104.69, 110.77].", "meta": {"id": "WUA_STAT_25K_009628", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.31, s=24.44, n=11, compute CI."}]}, "response": "95% CI ≈ [54.87, 83.75].", "meta": {"id": "WUA_STAT_25K_009629", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.46, compute E and Var."}]}, "response": "E[X]=18.40; Var[X]=9.936.", "meta": {"id": "WUA_STAT_25K_009630", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.56, compute E and Var."}]}, "response": "E[X]=21.84; Var[X]=9.610.", "meta": {"id": "WUA_STAT_25K_009631", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.093, sens=0.86, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.815.", "meta": {"id": "WUA_STAT_25K_009632", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.064, sens=0.73, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.200.", "meta": {"id": "WUA_STAT_25K_009633", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.014, sens=0.8, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.049.", "meta": {"id": "WUA_STAT_25K_009634", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.92, μ=75.29, σ=11.25, compute z."}]}, "response": "z = -0.655.", "meta": {"id": "WUA_STAT_25K_009635", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.141, sens=0.92, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.350.", "meta": {"id": "WUA_STAT_25K_009636", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=42.34, s=12.95, n=188, compute CI."}]}, "response": "95% CI ≈ [40.49, 44.19].", "meta": {"id": "WUA_STAT_25K_009637", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.6, μ=90.69, σ=19.98, compute z."}]}, "response": "z = 0.046.", "meta": {"id": "WUA_STAT_25K_009638", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.9, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.467.", "meta": {"id": "WUA_STAT_25K_009639", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.01, μ=88.95, σ=20.35, compute z."}]}, "response": "z = -3.240.", "meta": {"id": "WUA_STAT_25K_009640", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.84, s=37.04, n=53, compute CI."}]}, "response": "95% CI ≈ [33.87, 53.81].", "meta": {"id": "WUA_STAT_25K_009641", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.084, sens=0.78, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.338.", "meta": {"id": "WUA_STAT_25K_009642", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.08, s=3.82, n=154, compute CI."}]}, "response": "95% CI ≈ [100.48, 101.68].", "meta": {"id": "WUA_STAT_25K_009643", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.14, s=17.85, n=200, compute CI."}]}, "response": "95% CI ≈ [96.67, 101.61].", "meta": {"id": "WUA_STAT_25K_009644", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.46, μ=96.4, σ=21.03, compute z."}]}, "response": "z = -3.040.", "meta": {"id": "WUA_STAT_25K_009645", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.208, sens=0.79, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.509.", "meta": {"id": "WUA_STAT_25K_009646", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.15, s=1.08, n=26, compute CI."}]}, "response": "95% CI ≈ [66.73, 67.57].", "meta": {"id": "WUA_STAT_25K_009647", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.97, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.132.", "meta": {"id": "WUA_STAT_25K_009648", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.43, μ=79.89, σ=17.34, compute z."}]}, "response": "z = 0.377.", "meta": {"id": "WUA_STAT_25K_009649", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.76, compute E and Var."}]}, "response": "E[X]=11.40; Var[X]=2.736.", "meta": {"id": "WUA_STAT_25K_009650", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=197.22, s=13.75, n=78, compute CI."}]}, "response": "95% CI ≈ [194.17, 200.27].", "meta": {"id": "WUA_STAT_25K_009651", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.16, μ=73.2, σ=7.34, compute z."}]}, "response": "z = -3.684.", "meta": {"id": "WUA_STAT_25K_009652", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.27, s=28.16, n=83, compute CI."}]}, "response": "95% CI ≈ [73.21, 85.33].", "meta": {"id": "WUA_STAT_25K_009653", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.37, compute E and Var."}]}, "response": "E[X]=12.95; Var[X]=8.159.", "meta": {"id": "WUA_STAT_25K_009654", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.153, sens=0.89, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.517.", "meta": {"id": "WUA_STAT_25K_009655", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.93, μ=76.53, σ=6.84, compute z."}]}, "response": "z = 2.836.", "meta": {"id": "WUA_STAT_25K_009656", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.52, compute E and Var."}]}, "response": "E[X]=3.12; Var[X]=1.498.", "meta": {"id": "WUA_STAT_25K_009657", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.47, s=22.51, n=175, compute CI."}]}, "response": "95% CI ≈ [117.13, 123.81].", "meta": {"id": "WUA_STAT_25K_009658", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.65, compute E and Var."}]}, "response": "E[X]=8.45; Var[X]=2.958.", "meta": {"id": "WUA_STAT_25K_009659", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.167, sens=0.92, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.397.", "meta": {"id": "WUA_STAT_25K_009660", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.156, sens=0.73, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.600.", "meta": {"id": "WUA_STAT_25K_009661", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=108.23, s=1.4, n=153, compute CI."}]}, "response": "95% CI ≈ [108.01, 108.45].", "meta": {"id": "WUA_STAT_25K_009662", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.55, s=33.88, n=16, compute CI."}]}, "response": "95% CI ≈ [162.95, 196.15].", "meta": {"id": "WUA_STAT_25K_009663", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.089, sens=0.92, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.290.", "meta": {"id": "WUA_STAT_25K_009664", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.28, s=37.16, n=72, compute CI."}]}, "response": "95% CI ≈ [131.70, 148.86].", "meta": {"id": "WUA_STAT_25K_009665", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.72, μ=66.0, σ=21.58, compute z."}]}, "response": "z = -0.013.", "meta": {"id": "WUA_STAT_25K_009666", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.82, s=32.43, n=10, compute CI."}]}, "response": "95% CI ≈ [35.72, 75.92].", "meta": {"id": "WUA_STAT_25K_009667", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.24, s=8.51, n=170, compute CI."}]}, "response": "95% CI ≈ [96.96, 99.52].", "meta": {"id": "WUA_STAT_25K_009668", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.53, μ=7.21, σ=16.68, compute z."}]}, "response": "z = 3.796.", "meta": {"id": "WUA_STAT_25K_009669", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.91, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.669.", "meta": {"id": "WUA_STAT_25K_009670", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.28, s=30.98, n=161, compute CI."}]}, "response": "95% CI ≈ [180.49, 190.07].", "meta": {"id": "WUA_STAT_25K_009671", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.43, s=32.26, n=117, compute CI."}]}, "response": "95% CI ≈ [63.58, 75.28].", "meta": {"id": "WUA_STAT_25K_009672", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.04, s=21.44, n=117, compute CI."}]}, "response": "95% CI ≈ [171.16, 178.92].", "meta": {"id": "WUA_STAT_25K_009673", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.43, μ=96.82, σ=4.44, compute z."}]}, "response": "z = -3.917.", "meta": {"id": "WUA_STAT_25K_009674", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.97, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.132.", "meta": {"id": "WUA_STAT_25K_009675", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.67, compute E and Var."}]}, "response": "E[X]=12.06; Var[X]=3.980.", "meta": {"id": "WUA_STAT_25K_009676", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.1, compute E and Var."}]}, "response": "E[X]=2.50; Var[X]=2.250.", "meta": {"id": "WUA_STAT_25K_009677", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.74, μ=44.22, σ=18.26, compute z."}]}, "response": "z = 1.343.", "meta": {"id": "WUA_STAT_25K_009678", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.37, compute E and Var."}]}, "response": "E[X]=14.80; Var[X]=9.324.", "meta": {"id": "WUA_STAT_25K_009679", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.62, μ=85.41, σ=23.67, compute z."}]}, "response": "z = -2.357.", "meta": {"id": "WUA_STAT_25K_009680", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.12, μ=50.72, σ=11.67, compute z."}]}, "response": "z = -2.279.", "meta": {"id": "WUA_STAT_25K_009681", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.82, s=21.95, n=60, compute CI."}]}, "response": "95% CI ≈ [117.27, 128.37].", "meta": {"id": "WUA_STAT_25K_009682", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.83, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.508.", "meta": {"id": "WUA_STAT_25K_009683", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.39, compute E and Var."}]}, "response": "E[X]=9.75; Var[X]=5.947.", "meta": {"id": "WUA_STAT_25K_009684", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.72, s=13.03, n=165, compute CI."}]}, "response": "95% CI ≈ [111.73, 115.71].", "meta": {"id": "WUA_STAT_25K_009685", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.99, μ=21.45, σ=10.0, compute z."}]}, "response": "z = -1.146.", "meta": {"id": "WUA_STAT_25K_009686", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.216, sens=0.93, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.740.", "meta": {"id": "WUA_STAT_25K_009687", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.53, s=33.81, n=197, compute CI."}]}, "response": "95% CI ≈ [83.81, 93.25].", "meta": {"id": "WUA_STAT_25K_009688", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.85, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.559.", "meta": {"id": "WUA_STAT_25K_009689", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.45, compute E and Var."}]}, "response": "E[X]=16.65; Var[X]=9.158.", "meta": {"id": "WUA_STAT_25K_009690", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.12, compute E and Var."}]}, "response": "E[X]=3.72; Var[X]=3.274.", "meta": {"id": "WUA_STAT_25K_009691", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.21, sens=0.77, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.482.", "meta": {"id": "WUA_STAT_25K_009692", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.08, μ=18.72, σ=9.02, compute z."}]}, "response": "z = 7.246.", "meta": {"id": "WUA_STAT_25K_009693", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.118, sens=0.96, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.478.", "meta": {"id": "WUA_STAT_25K_009694", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.95, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.498.", "meta": {"id": "WUA_STAT_25K_009695", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.44, μ=88.15, σ=1.5, compute z."}]}, "response": "z = 0.860.", "meta": {"id": "WUA_STAT_25K_009696", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.022, sens=0.8, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.310.", "meta": {"id": "WUA_STAT_25K_009697", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.94, μ=41.04, σ=22.32, compute z."}]}, "response": "z = 0.533.", "meta": {"id": "WUA_STAT_25K_009698", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.214, sens=0.83, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.496.", "meta": {"id": "WUA_STAT_25K_009699", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.21, s=25.32, n=150, compute CI."}]}, "response": "95% CI ≈ [63.16, 71.26].", "meta": {"id": "WUA_STAT_25K_009700", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.19, s=12.81, n=109, compute CI."}]}, "response": "95% CI ≈ [14.79, 19.59].", "meta": {"id": "WUA_STAT_25K_009701", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.48, compute E and Var."}]}, "response": "E[X]=14.88; Var[X]=7.738.", "meta": {"id": "WUA_STAT_25K_009702", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.05, μ=11.63, σ=13.18, compute z."}]}, "response": "z = 1.322.", "meta": {"id": "WUA_STAT_25K_009703", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.63, compute E and Var."}]}, "response": "E[X]=13.86; Var[X]=5.128.", "meta": {"id": "WUA_STAT_25K_009704", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.115, sens=0.98, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.305.", "meta": {"id": "WUA_STAT_25K_009705", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=56.83, s=28.93, n=82, compute CI."}]}, "response": "95% CI ≈ [50.57, 63.09].", "meta": {"id": "WUA_STAT_25K_009706", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.82, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.622.", "meta": {"id": "WUA_STAT_25K_009707", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.43, compute E and Var."}]}, "response": "E[X]=8.17; Var[X]=4.657.", "meta": {"id": "WUA_STAT_25K_009708", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.99, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.279.", "meta": {"id": "WUA_STAT_25K_009709", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.98, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.841.", "meta": {"id": "WUA_STAT_25K_009710", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.75, compute E and Var."}]}, "response": "E[X]=27.75; Var[X]=6.938.", "meta": {"id": "WUA_STAT_25K_009711", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.135, sens=0.97, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.368.", "meta": {"id": "WUA_STAT_25K_009712", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.08, μ=51.46, σ=18.97, compute z."}]}, "response": "z = -0.125.", "meta": {"id": "WUA_STAT_25K_009713", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.6, μ=83.78, σ=23.28, compute z."}]}, "response": "z = -1.253.", "meta": {"id": "WUA_STAT_25K_009714", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.24, sens=0.92, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.580.", "meta": {"id": "WUA_STAT_25K_009715", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.86, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.783.", "meta": {"id": "WUA_STAT_25K_009716", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.45, s=23.39, n=49, compute CI."}]}, "response": "95% CI ≈ [10.90, 24.00].", "meta": {"id": "WUA_STAT_25K_009717", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.71, compute E and Var."}]}, "response": "E[X]=23.43; Var[X]=6.795.", "meta": {"id": "WUA_STAT_25K_009718", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.52, s=13.66, n=32, compute CI."}]}, "response": "95% CI ≈ [85.79, 95.25].", "meta": {"id": "WUA_STAT_25K_009719", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.077, sens=0.89, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.229.", "meta": {"id": "WUA_STAT_25K_009720", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.28, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=4.032.", "meta": {"id": "WUA_STAT_25K_009721", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.197, sens=0.92, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.715.", "meta": {"id": "WUA_STAT_25K_009722", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.22, s=25.47, n=37, compute CI."}]}, "response": "95% CI ≈ [57.01, 73.43].", "meta": {"id": "WUA_STAT_25K_009723", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.3, compute E and Var."}]}, "response": "E[X]=2.70; Var[X]=1.890.", "meta": {"id": "WUA_STAT_25K_009724", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.71, μ=5.3, σ=22.88, compute z."}]}, "response": "z = 3.340.", "meta": {"id": "WUA_STAT_25K_009725", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.12, compute E and Var."}]}, "response": "E[X]=2.16; Var[X]=1.901.", "meta": {"id": "WUA_STAT_25K_009726", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.95, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.611.", "meta": {"id": "WUA_STAT_25K_009727", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.201, sens=0.78, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.641.", "meta": {"id": "WUA_STAT_25K_009728", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.125, sens=0.85, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.327.", "meta": {"id": "WUA_STAT_25K_009729", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.57, compute E and Var."}]}, "response": "E[X]=6.27; Var[X]=2.696.", "meta": {"id": "WUA_STAT_25K_009730", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.25, μ=14.2, σ=2.95, compute z."}]}, "response": "z = 4.763.", "meta": {"id": "WUA_STAT_25K_009731", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.82, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.740.", "meta": {"id": "WUA_STAT_25K_009732", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.27, μ=60.73, σ=3.55, compute z."}]}, "response": "z = 4.941.", "meta": {"id": "WUA_STAT_25K_009733", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.68, s=13.28, n=146, compute CI."}]}, "response": "95% CI ≈ [11.53, 15.83].", "meta": {"id": "WUA_STAT_25K_009734", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.228, sens=0.98, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.591.", "meta": {"id": "WUA_STAT_25K_009735", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.71, μ=3.36, σ=19.55, compute z."}]}, "response": "z = 3.803.", "meta": {"id": "WUA_STAT_25K_009736", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.98, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.259.", "meta": {"id": "WUA_STAT_25K_009737", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.12, s=3.37, n=30, compute CI."}]}, "response": "95% CI ≈ [86.91, 89.33].", "meta": {"id": "WUA_STAT_25K_009738", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.43, compute E and Var."}]}, "response": "E[X]=9.89; Var[X]=5.637.", "meta": {"id": "WUA_STAT_25K_009739", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.46, compute E and Var."}]}, "response": "E[X]=9.20; Var[X]=4.968.", "meta": {"id": "WUA_STAT_25K_009740", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.246, sens=0.84, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.646.", "meta": {"id": "WUA_STAT_25K_009741", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.32, compute E and Var."}]}, "response": "E[X]=9.92; Var[X]=6.746.", "meta": {"id": "WUA_STAT_25K_009742", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.28, compute E and Var."}]}, "response": "E[X]=8.12; Var[X]=5.846.", "meta": {"id": "WUA_STAT_25K_009743", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.46, compute E and Var."}]}, "response": "E[X]=3.68; Var[X]=1.987.", "meta": {"id": "WUA_STAT_25K_009744", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.42, compute E and Var."}]}, "response": "E[X]=4.62; Var[X]=2.680.", "meta": {"id": "WUA_STAT_25K_009745", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.75, s=22.98, n=31, compute CI."}]}, "response": "95% CI ≈ [55.66, 71.84].", "meta": {"id": "WUA_STAT_25K_009746", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.15, μ=63.95, σ=20.01, compute z."}]}, "response": "z = -2.339.", "meta": {"id": "WUA_STAT_25K_009747", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.62, compute E and Var."}]}, "response": "E[X]=19.22; Var[X]=7.304.", "meta": {"id": "WUA_STAT_25K_009748", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.1, compute E and Var."}]}, "response": "E[X]=1.10; Var[X]=0.990.", "meta": {"id": "WUA_STAT_25K_009749", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.52, μ=12.21, σ=2.61, compute z."}]}, "response": "z = 23.490.", "meta": {"id": "WUA_STAT_25K_009750", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.99, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.895.", "meta": {"id": "WUA_STAT_25K_009751", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.91, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.593.", "meta": {"id": "WUA_STAT_25K_009752", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.83, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.404.", "meta": {"id": "WUA_STAT_25K_009753", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.13, μ=10.19, σ=14.6, compute z."}]}, "response": "z = 3.078.", "meta": {"id": "WUA_STAT_25K_009754", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.6, compute E and Var."}]}, "response": "E[X]=6.00; Var[X]=2.400.", "meta": {"id": "WUA_STAT_25K_009755", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.3, compute E and Var."}]}, "response": "E[X]=8.10; Var[X]=5.670.", "meta": {"id": "WUA_STAT_25K_009756", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.55, s=19.08, n=99, compute CI."}]}, "response": "95% CI ≈ [56.79, 64.31].", "meta": {"id": "WUA_STAT_25K_009757", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.223, sens=0.95, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.602.", "meta": {"id": "WUA_STAT_25K_009758", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.7, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.495.", "meta": {"id": "WUA_STAT_25K_009759", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=66.03, s=12.96, n=52, compute CI."}]}, "response": "95% CI ≈ [62.51, 69.55].", "meta": {"id": "WUA_STAT_25K_009760", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.78, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.708.", "meta": {"id": "WUA_STAT_25K_009761", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.06, μ=35.44, σ=16.73, compute z."}]}, "response": "z = 1.352.", "meta": {"id": "WUA_STAT_25K_009762", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.09, sens=0.95, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.824.", "meta": {"id": "WUA_STAT_25K_009763", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=51.23, s=27.21, n=170, compute CI."}]}, "response": "95% CI ≈ [47.14, 55.32].", "meta": {"id": "WUA_STAT_25K_009764", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.45, compute E and Var."}]}, "response": "E[X]=10.80; Var[X]=5.940.", "meta": {"id": "WUA_STAT_25K_009765", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.88, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.945.", "meta": {"id": "WUA_STAT_25K_009766", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.41, μ=79.28, σ=4.02, compute z."}]}, "response": "z = -17.132.", "meta": {"id": "WUA_STAT_25K_009767", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.29, μ=45.77, σ=16.62, compute z."}]}, "response": "z = 0.813.", "meta": {"id": "WUA_STAT_25K_009768", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.31, compute E and Var."}]}, "response": "E[X]=4.96; Var[X]=3.422.", "meta": {"id": "WUA_STAT_25K_009769", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.64, μ=5.36, σ=5.57, compute z."}]}, "response": "z = 14.233.", "meta": {"id": "WUA_STAT_25K_009770", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.3, μ=84.57, σ=3.64, compute z."}]}, "response": "z = 3.772.", "meta": {"id": "WUA_STAT_25K_009771", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.7, μ=98.32, σ=8.33, compute z."}]}, "response": "z = -0.315.", "meta": {"id": "WUA_STAT_25K_009772", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.23, compute E and Var."}]}, "response": "E[X]=5.06; Var[X]=3.896.", "meta": {"id": "WUA_STAT_25K_009773", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.78, s=6.07, n=123, compute CI."}]}, "response": "95% CI ≈ [137.71, 139.85].", "meta": {"id": "WUA_STAT_25K_009774", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.15, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.785.", "meta": {"id": "WUA_STAT_25K_009775", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.45, compute E and Var."}]}, "response": "E[X]=8.55; Var[X]=4.703.", "meta": {"id": "WUA_STAT_25K_009776", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.019, sens=0.84, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.140.", "meta": {"id": "WUA_STAT_25K_009777", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.97, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.293.", "meta": {"id": "WUA_STAT_25K_009778", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.55, μ=23.98, σ=13.43, compute z."}]}, "response": "z = -1.596.", "meta": {"id": "WUA_STAT_25K_009779", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.2, compute E and Var."}]}, "response": "E[X]=4.60; Var[X]=3.680.", "meta": {"id": "WUA_STAT_25K_009780", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.75, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.279.", "meta": {"id": "WUA_STAT_25K_009781", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.084, sens=0.92, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.738.", "meta": {"id": "WUA_STAT_25K_009782", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.37, s=15.84, n=65, compute CI."}]}, "response": "95% CI ≈ [186.52, 194.22].", "meta": {"id": "WUA_STAT_25K_009783", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=73.89, s=16.7, n=43, compute CI."}]}, "response": "95% CI ≈ [68.90, 78.88].", "meta": {"id": "WUA_STAT_25K_009784", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.24, sens=0.87, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.504.", "meta": {"id": "WUA_STAT_25K_009785", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.86, μ=65.84, σ=11.62, compute z."}]}, "response": "z = -4.043.", "meta": {"id": "WUA_STAT_25K_009786", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=96.15, s=3.9, n=149, compute CI."}]}, "response": "95% CI ≈ [95.52, 96.78].", "meta": {"id": "WUA_STAT_25K_009787", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.92, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.387.", "meta": {"id": "WUA_STAT_25K_009788", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.75, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.202.", "meta": {"id": "WUA_STAT_25K_009789", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.92, s=13.69, n=185, compute CI."}]}, "response": "95% CI ≈ [176.95, 180.89].", "meta": {"id": "WUA_STAT_25K_009790", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.45, compute E and Var."}]}, "response": "E[X]=18.00; Var[X]=9.900.", "meta": {"id": "WUA_STAT_25K_009791", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.86, s=17.54, n=27, compute CI."}]}, "response": "95% CI ≈ [91.24, 104.48].", "meta": {"id": "WUA_STAT_25K_009792", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.115, sens=0.96, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.324.", "meta": {"id": "WUA_STAT_25K_009793", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.2, μ=32.69, σ=21.79, compute z."}]}, "response": "z = -0.757.", "meta": {"id": "WUA_STAT_25K_009794", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.43, compute E and Var."}]}, "response": "E[X]=15.48; Var[X]=8.824.", "meta": {"id": "WUA_STAT_25K_009795", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.76, s=14.07, n=139, compute CI."}]}, "response": "95% CI ≈ [185.42, 190.10].", "meta": {"id": "WUA_STAT_25K_009796", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.17, compute E and Var."}]}, "response": "E[X]=4.08; Var[X]=3.386.", "meta": {"id": "WUA_STAT_25K_009797", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.72, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.235.", "meta": {"id": "WUA_STAT_25K_009798", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.67, μ=21.43, σ=18.18, compute z."}]}, "response": "z = -0.262.", "meta": {"id": "WUA_STAT_25K_009799", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.32, μ=91.21, σ=7.73, compute z."}]}, "response": "z = -2.961.", "meta": {"id": "WUA_STAT_25K_009800", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.52, compute E and Var."}]}, "response": "E[X]=9.88; Var[X]=4.742.", "meta": {"id": "WUA_STAT_25K_009801", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.39, μ=51.4, σ=15.99, compute z."}]}, "response": "z = 0.625.", "meta": {"id": "WUA_STAT_25K_009802", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.89, μ=29.75, σ=19.09, compute z."}]}, "response": "z = 3.674.", "meta": {"id": "WUA_STAT_25K_009803", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.25, sens=0.95, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.841.", "meta": {"id": "WUA_STAT_25K_009804", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.18, μ=43.07, σ=15.58, compute z."}]}, "response": "z = -1.212.", "meta": {"id": "WUA_STAT_25K_009805", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.72, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.791.", "meta": {"id": "WUA_STAT_25K_009806", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.08, s=12.16, n=68, compute CI."}]}, "response": "95% CI ≈ [20.19, 25.97].", "meta": {"id": "WUA_STAT_25K_009807", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.73, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.577.", "meta": {"id": "WUA_STAT_25K_009808", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.29, compute E and Var."}]}, "response": "E[X]=5.80; Var[X]=4.118.", "meta": {"id": "WUA_STAT_25K_009809", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.12, compute E and Var."}]}, "response": "E[X]=1.92; Var[X]=1.690.", "meta": {"id": "WUA_STAT_25K_009810", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.29, compute E and Var."}]}, "response": "E[X]=4.93; Var[X]=3.500.", "meta": {"id": "WUA_STAT_25K_009811", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.03, μ=66.58, σ=3.49, compute z."}]}, "response": "z = 8.152.", "meta": {"id": "WUA_STAT_25K_009812", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.89, μ=86.53, σ=2.29, compute z."}]}, "response": "z = 2.777.", "meta": {"id": "WUA_STAT_25K_009813", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.89, μ=58.36, σ=14.77, compute z."}]}, "response": "z = -2.063.", "meta": {"id": "WUA_STAT_25K_009814", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.36, μ=71.25, σ=1.47, compute z."}]}, "response": "z = -12.850.", "meta": {"id": "WUA_STAT_25K_009815", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.56, compute E and Var."}]}, "response": "E[X]=4.48; Var[X]=1.971.", "meta": {"id": "WUA_STAT_25K_009816", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.02, sens=0.83, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.361.", "meta": {"id": "WUA_STAT_25K_009817", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.43, s=30.24, n=135, compute CI."}]}, "response": "95% CI ≈ [151.33, 161.53].", "meta": {"id": "WUA_STAT_25K_009818", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.82, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.540.", "meta": {"id": "WUA_STAT_25K_009819", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.32, compute E and Var."}]}, "response": "E[X]=7.36; Var[X]=5.005.", "meta": {"id": "WUA_STAT_25K_009820", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.72, compute E and Var."}]}, "response": "E[X]=16.56; Var[X]=4.637.", "meta": {"id": "WUA_STAT_25K_009821", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.84, μ=97.98, σ=6.85, compute z."}]}, "response": "z = -13.889.", "meta": {"id": "WUA_STAT_25K_009822", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.71, compute E and Var."}]}, "response": "E[X]=19.88; Var[X]=5.765.", "meta": {"id": "WUA_STAT_25K_009823", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.6, compute E and Var."}]}, "response": "E[X]=9.60; Var[X]=3.840.", "meta": {"id": "WUA_STAT_25K_009824", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.81, s=37.17, n=38, compute CI."}]}, "response": "95% CI ≈ [154.99, 178.63].", "meta": {"id": "WUA_STAT_25K_009825", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.19, compute E and Var."}]}, "response": "E[X]=2.66; Var[X]=2.155.", "meta": {"id": "WUA_STAT_25K_009826", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.22, μ=91.24, σ=14.8, compute z."}]}, "response": "z = -2.028.", "meta": {"id": "WUA_STAT_25K_009827", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.58, compute E and Var."}]}, "response": "E[X]=23.20; Var[X]=9.744.", "meta": {"id": "WUA_STAT_25K_009828", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.2, compute E and Var."}]}, "response": "E[X]=1.40; Var[X]=1.120.", "meta": {"id": "WUA_STAT_25K_009829", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.3, compute E and Var."}]}, "response": "E[X]=6.30; Var[X]=4.410.", "meta": {"id": "WUA_STAT_25K_009830", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.05, μ=65.52, σ=20.93, compute z."}]}, "response": "z = -0.309.", "meta": {"id": "WUA_STAT_25K_009831", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.15, compute E and Var."}]}, "response": "E[X]=3.30; Var[X]=2.805.", "meta": {"id": "WUA_STAT_25K_009832", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.77, μ=74.9, σ=18.25, compute z."}]}, "response": "z = -3.404.", "meta": {"id": "WUA_STAT_25K_009833", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.84, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.391.", "meta": {"id": "WUA_STAT_25K_009834", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.96, s=27.34, n=78, compute CI."}]}, "response": "95% CI ≈ [17.89, 30.03].", "meta": {"id": "WUA_STAT_25K_009835", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.71, s=9.88, n=72, compute CI."}]}, "response": "95% CI ≈ [48.43, 52.99].", "meta": {"id": "WUA_STAT_25K_009836", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.231, sens=0.73, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.458.", "meta": {"id": "WUA_STAT_25K_009837", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.67, compute E and Var."}]}, "response": "E[X]=9.38; Var[X]=3.095.", "meta": {"id": "WUA_STAT_25K_009838", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.39, s=31.89, n=33, compute CI."}]}, "response": "95% CI ≈ [187.51, 209.27].", "meta": {"id": "WUA_STAT_25K_009839", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.99, μ=41.52, σ=3.61, compute z."}]}, "response": "z = 2.346.", "meta": {"id": "WUA_STAT_25K_009840", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.67, μ=82.79, σ=20.23, compute z."}]}, "response": "z = -0.995.", "meta": {"id": "WUA_STAT_25K_009841", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.82, s=12.84, n=181, compute CI."}]}, "response": "95% CI ≈ [47.95, 51.69].", "meta": {"id": "WUA_STAT_25K_009842", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.06, compute E and Var."}]}, "response": "E[X]=1.50; Var[X]=1.410.", "meta": {"id": "WUA_STAT_25K_009843", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.36, s=26.99, n=97, compute CI."}]}, "response": "95% CI ≈ [48.99, 59.73].", "meta": {"id": "WUA_STAT_25K_009844", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.91, μ=2.11, σ=22.36, compute z."}]}, "response": "z = 3.390.", "meta": {"id": "WUA_STAT_25K_009845", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.165, sens=0.84, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.648.", "meta": {"id": "WUA_STAT_25K_009846", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.21, μ=84.24, σ=14.44, compute z."}]}, "response": "z = -0.695.", "meta": {"id": "WUA_STAT_25K_009847", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.146, sens=0.93, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.550.", "meta": {"id": "WUA_STAT_25K_009848", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.41, compute E and Var."}]}, "response": "E[X]=12.30; Var[X]=7.257.", "meta": {"id": "WUA_STAT_25K_009849", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.42, s=38.95, n=167, compute CI."}]}, "response": "95% CI ≈ [101.51, 113.33].", "meta": {"id": "WUA_STAT_25K_009850", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.28, compute E and Var."}]}, "response": "E[X]=6.72; Var[X]=4.838.", "meta": {"id": "WUA_STAT_25K_009851", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.53, μ=55.54, σ=19.52, compute z."}]}, "response": "z = -1.589.", "meta": {"id": "WUA_STAT_25K_009852", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.64, s=2.11, n=11, compute CI."}]}, "response": "95% CI ≈ [139.39, 141.89].", "meta": {"id": "WUA_STAT_25K_009853", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.24, μ=22.04, σ=15.73, compute z."}]}, "response": "z = 1.221.", "meta": {"id": "WUA_STAT_25K_009854", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.32, μ=41.41, σ=9.81, compute z."}]}, "response": "z = 3.151.", "meta": {"id": "WUA_STAT_25K_009855", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.73, compute E and Var."}]}, "response": "E[X]=24.09; Var[X]=6.504.", "meta": {"id": "WUA_STAT_25K_009856", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.84, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.374.", "meta": {"id": "WUA_STAT_25K_009857", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.88, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.370.", "meta": {"id": "WUA_STAT_25K_009858", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.29, μ=59.3, σ=14.68, compute z."}]}, "response": "z = -2.521.", "meta": {"id": "WUA_STAT_25K_009859", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.92, s=35.72, n=92, compute CI."}]}, "response": "95% CI ≈ [8.62, 23.22].", "meta": {"id": "WUA_STAT_25K_009860", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.17, s=33.89, n=60, compute CI."}]}, "response": "95% CI ≈ [89.59, 106.75].", "meta": {"id": "WUA_STAT_25K_009861", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.179, sens=0.73, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.888.", "meta": {"id": "WUA_STAT_25K_009862", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.43, compute E and Var."}]}, "response": "E[X]=15.05; Var[X]=8.579.", "meta": {"id": "WUA_STAT_25K_009863", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.066, sens=0.85, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.353.", "meta": {"id": "WUA_STAT_25K_009864", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.46, s=20.39, n=187, compute CI."}]}, "response": "95% CI ≈ [179.54, 185.38].", "meta": {"id": "WUA_STAT_25K_009865", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.95, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.256.", "meta": {"id": "WUA_STAT_25K_009866", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.9, μ=63.57, σ=18.09, compute z."}]}, "response": "z = -2.967.", "meta": {"id": "WUA_STAT_25K_009867", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.78, compute E and Var."}]}, "response": "E[X]=31.20; Var[X]=6.864.", "meta": {"id": "WUA_STAT_25K_009868", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.43, compute E and Var."}]}, "response": "E[X]=15.05; Var[X]=8.579.", "meta": {"id": "WUA_STAT_25K_009869", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.45, s=29.46, n=175, compute CI."}]}, "response": "95% CI ≈ [162.09, 170.81].", "meta": {"id": "WUA_STAT_25K_009870", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.38, μ=13.87, σ=8.13, compute z."}]}, "response": "z = -0.306.", "meta": {"id": "WUA_STAT_25K_009871", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.149, sens=0.81, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.563.", "meta": {"id": "WUA_STAT_25K_009872", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.99, s=36.26, n=115, compute CI."}]}, "response": "95% CI ≈ [49.36, 62.62].", "meta": {"id": "WUA_STAT_25K_009873", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.0, s=23.52, n=123, compute CI."}]}, "response": "95% CI ≈ [115.84, 124.16].", "meta": {"id": "WUA_STAT_25K_009874", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.76, s=32.69, n=167, compute CI."}]}, "response": "95% CI ≈ [133.80, 143.72].", "meta": {"id": "WUA_STAT_25K_009875", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.67, s=35.13, n=90, compute CI."}]}, "response": "95% CI ≈ [27.41, 41.93].", "meta": {"id": "WUA_STAT_25K_009876", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.156, sens=0.76, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.824.", "meta": {"id": "WUA_STAT_25K_009877", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=183.36, s=24.83, n=47, compute CI."}]}, "response": "95% CI ≈ [176.26, 190.46].", "meta": {"id": "WUA_STAT_25K_009878", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=51.07, s=21.01, n=115, compute CI."}]}, "response": "95% CI ≈ [47.23, 54.91].", "meta": {"id": "WUA_STAT_25K_009879", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.49, compute E and Var."}]}, "response": "E[X]=13.23; Var[X]=6.747.", "meta": {"id": "WUA_STAT_25K_009880", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.67, compute E and Var."}]}, "response": "E[X]=10.05; Var[X]=3.317.", "meta": {"id": "WUA_STAT_25K_009881", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.42, μ=70.87, σ=2.91, compute z."}]}, "response": "z = -2.904.", "meta": {"id": "WUA_STAT_25K_009882", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.043, sens=0.72, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.111.", "meta": {"id": "WUA_STAT_25K_009883", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.195, sens=0.91, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.595.", "meta": {"id": "WUA_STAT_25K_009884", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.16, s=2.03, n=158, compute CI."}]}, "response": "95% CI ≈ [133.84, 134.48].", "meta": {"id": "WUA_STAT_25K_009885", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.81, s=33.51, n=187, compute CI."}]}, "response": "95% CI ≈ [18.01, 27.61].", "meta": {"id": "WUA_STAT_25K_009886", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.56, μ=92.55, σ=12.61, compute z."}]}, "response": "z = -5.709.", "meta": {"id": "WUA_STAT_25K_009887", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.77, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.844.", "meta": {"id": "WUA_STAT_25K_009888", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.38, μ=94.54, σ=7.29, compute z."}]}, "response": "z = -7.292.", "meta": {"id": "WUA_STAT_25K_009889", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=16.75, s=28.95, n=80, compute CI."}]}, "response": "95% CI ≈ [10.41, 23.09].", "meta": {"id": "WUA_STAT_25K_009890", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.78, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.366.", "meta": {"id": "WUA_STAT_25K_009891", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.96, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.504.", "meta": {"id": "WUA_STAT_25K_009892", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.86, s=24.53, n=186, compute CI."}]}, "response": "95% CI ≈ [129.33, 136.39].", "meta": {"id": "WUA_STAT_25K_009893", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.45, compute E and Var."}]}, "response": "E[X]=4.05; Var[X]=2.228.", "meta": {"id": "WUA_STAT_25K_009894", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.77, compute E and Var."}]}, "response": "E[X]=22.33; Var[X]=5.136.", "meta": {"id": "WUA_STAT_25K_009895", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.53, s=34.91, n=103, compute CI."}]}, "response": "95% CI ≈ [131.79, 145.27].", "meta": {"id": "WUA_STAT_25K_009896", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.54, s=25.44, n=183, compute CI."}]}, "response": "95% CI ≈ [123.85, 131.23].", "meta": {"id": "WUA_STAT_25K_009897", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.241, sens=0.95, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.883.", "meta": {"id": "WUA_STAT_25K_009898", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.69, μ=76.42, σ=24.84, compute z."}]}, "response": "z = 0.655.", "meta": {"id": "WUA_STAT_25K_009899", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.82, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.883.", "meta": {"id": "WUA_STAT_25K_009900", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.74, μ=19.77, σ=1.52, compute z."}]}, "response": "z = 21.033.", "meta": {"id": "WUA_STAT_25K_009901", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.61, μ=45.53, σ=17.26, compute z."}]}, "response": "z = -1.444.", "meta": {"id": "WUA_STAT_25K_009902", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.12, μ=71.26, σ=22.86, compute z."}]}, "response": "z = -1.318.", "meta": {"id": "WUA_STAT_25K_009903", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.85, μ=49.86, σ=10.92, compute z."}]}, "response": "z = -3.939.", "meta": {"id": "WUA_STAT_25K_009904", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.37, s=10.13, n=25, compute CI."}]}, "response": "95% CI ≈ [76.40, 84.34].", "meta": {"id": "WUA_STAT_25K_009905", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.18, μ=39.75, σ=1.53, compute z."}]}, "response": "z = -12.791.", "meta": {"id": "WUA_STAT_25K_009906", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.25, compute E and Var."}]}, "response": "E[X]=2.00; Var[X]=1.500.", "meta": {"id": "WUA_STAT_25K_009907", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.24, compute E and Var."}]}, "response": "E[X]=3.36; Var[X]=2.554.", "meta": {"id": "WUA_STAT_25K_009908", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.64, s=19.13, n=151, compute CI."}]}, "response": "95% CI ≈ [169.59, 175.69].", "meta": {"id": "WUA_STAT_25K_009909", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.35, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=3.640.", "meta": {"id": "WUA_STAT_25K_009910", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.41, compute E and Var."}]}, "response": "E[X]=8.61; Var[X]=5.080.", "meta": {"id": "WUA_STAT_25K_009911", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.71, s=24.28, n=84, compute CI."}]}, "response": "95% CI ≈ [185.52, 195.90].", "meta": {"id": "WUA_STAT_25K_009912", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.057, sens=0.85, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.160.", "meta": {"id": "WUA_STAT_25K_009913", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.98, μ=84.01, σ=24.94, compute z."}]}, "response": "z = -3.249.", "meta": {"id": "WUA_STAT_25K_009914", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.22, μ=12.31, σ=23.38, compute z."}]}, "response": "z = 1.750.", "meta": {"id": "WUA_STAT_25K_009915", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.49, μ=1.76, σ=2.59, compute z."}]}, "response": "z = 3.757.", "meta": {"id": "WUA_STAT_25K_009916", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.36, μ=94.79, σ=20.09, compute z."}]}, "response": "z = -2.809.", "meta": {"id": "WUA_STAT_25K_009917", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.77, compute E and Var."}]}, "response": "E[X]=12.32; Var[X]=2.834.", "meta": {"id": "WUA_STAT_25K_009918", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.86, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.438.", "meta": {"id": "WUA_STAT_25K_009919", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.78, s=19.04, n=54, compute CI."}]}, "response": "95% CI ≈ [40.70, 50.86].", "meta": {"id": "WUA_STAT_25K_009920", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.34, s=3.63, n=170, compute CI."}]}, "response": "95% CI ≈ [151.79, 152.89].", "meta": {"id": "WUA_STAT_25K_009921", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.19, compute E and Var."}]}, "response": "E[X]=6.46; Var[X]=5.233.", "meta": {"id": "WUA_STAT_25K_009922", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.63, μ=68.06, σ=11.9, compute z."}]}, "response": "z = -3.313.", "meta": {"id": "WUA_STAT_25K_009923", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.11, s=36.45, n=50, compute CI."}]}, "response": "95% CI ≈ [57.01, 77.21].", "meta": {"id": "WUA_STAT_25K_009924", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.9, μ=61.11, σ=2.1, compute z."}]}, "response": "z = -23.910.", "meta": {"id": "WUA_STAT_25K_009925", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.3, μ=53.2, σ=4.62, compute z."}]}, "response": "z = -6.039.", "meta": {"id": "WUA_STAT_25K_009926", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.153, sens=0.81, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.785.", "meta": {"id": "WUA_STAT_25K_009927", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.52, μ=20.47, σ=7.19, compute z."}]}, "response": "z = -0.132.", "meta": {"id": "WUA_STAT_25K_009928", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.8, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.861.", "meta": {"id": "WUA_STAT_25K_009929", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.043, sens=0.76, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.105.", "meta": {"id": "WUA_STAT_25K_009930", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.07, compute E and Var."}]}, "response": "E[X]=0.42; Var[X]=0.391.", "meta": {"id": "WUA_STAT_25K_009931", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.17, μ=6.45, σ=8.0, compute z."}]}, "response": "z = 7.840.", "meta": {"id": "WUA_STAT_25K_009932", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.96, μ=74.59, σ=16.82, compute z."}]}, "response": "z = -2.059.", "meta": {"id": "WUA_STAT_25K_009933", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.237, sens=0.86, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.517.", "meta": {"id": "WUA_STAT_25K_009934", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.63, μ=32.19, σ=20.0, compute z."}]}, "response": "z = 0.872.", "meta": {"id": "WUA_STAT_25K_009935", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.176, sens=0.74, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.397.", "meta": {"id": "WUA_STAT_25K_009936", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.53, compute E and Var."}]}, "response": "E[X]=10.60; Var[X]=4.982.", "meta": {"id": "WUA_STAT_25K_009937", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.04, μ=87.3, σ=11.77, compute z."}]}, "response": "z = -0.787.", "meta": {"id": "WUA_STAT_25K_009938", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.93, μ=18.29, σ=3.53, compute z."}]}, "response": "z = -0.669.", "meta": {"id": "WUA_STAT_25K_009939", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.168, sens=0.71, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.878.", "meta": {"id": "WUA_STAT_25K_009940", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.225, sens=0.76, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.424.", "meta": {"id": "WUA_STAT_25K_009941", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.43, s=9.31, n=25, compute CI."}]}, "response": "95% CI ≈ [127.78, 135.08].", "meta": {"id": "WUA_STAT_25K_009942", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.79, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.207.", "meta": {"id": "WUA_STAT_25K_009943", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.42, compute E and Var."}]}, "response": "E[X]=8.82; Var[X]=5.116.", "meta": {"id": "WUA_STAT_25K_009944", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=183.88, s=5.62, n=76, compute CI."}]}, "response": "95% CI ≈ [182.62, 185.14].", "meta": {"id": "WUA_STAT_25K_009945", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.095, sens=0.74, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.261.", "meta": {"id": "WUA_STAT_25K_009946", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.61, s=26.38, n=158, compute CI."}]}, "response": "95% CI ≈ [110.50, 118.72].", "meta": {"id": "WUA_STAT_25K_009947", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.072, sens=0.79, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.320.", "meta": {"id": "WUA_STAT_25K_009948", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.19, compute E and Var."}]}, "response": "E[X]=4.37; Var[X]=3.540.", "meta": {"id": "WUA_STAT_25K_009949", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.3, compute E and Var."}]}, "response": "E[X]=10.20; Var[X]=7.140.", "meta": {"id": "WUA_STAT_25K_009950", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.97, s=34.33, n=32, compute CI."}]}, "response": "95% CI ≈ [67.08, 90.86].", "meta": {"id": "WUA_STAT_25K_009951", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.81, s=1.73, n=39, compute CI."}]}, "response": "95% CI ≈ [36.27, 37.35].", "meta": {"id": "WUA_STAT_25K_009952", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.35, compute E and Var."}]}, "response": "E[X]=3.50; Var[X]=2.275.", "meta": {"id": "WUA_STAT_25K_009953", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.91, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.292.", "meta": {"id": "WUA_STAT_25K_009954", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.79, compute E and Var."}]}, "response": "E[X]=5.53; Var[X]=1.161.", "meta": {"id": "WUA_STAT_25K_009955", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.06, compute E and Var."}]}, "response": "E[X]=1.38; Var[X]=1.297.", "meta": {"id": "WUA_STAT_25K_009956", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=171.12, s=5.94, n=92, compute CI."}]}, "response": "95% CI ≈ [169.91, 172.33].", "meta": {"id": "WUA_STAT_25K_009957", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.47, compute E and Var."}]}, "response": "E[X]=15.04; Var[X]=7.971.", "meta": {"id": "WUA_STAT_25K_009958", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.033, sens=0.9, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.161.", "meta": {"id": "WUA_STAT_25K_009959", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.91, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.417.", "meta": {"id": "WUA_STAT_25K_009960", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.12, compute E and Var."}]}, "response": "E[X]=0.96; Var[X]=0.845.", "meta": {"id": "WUA_STAT_25K_009961", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.97, s=5.53, n=43, compute CI."}]}, "response": "95% CI ≈ [60.32, 63.62].", "meta": {"id": "WUA_STAT_25K_009962", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.94, μ=26.29, σ=4.85, compute z."}]}, "response": "z = -0.072.", "meta": {"id": "WUA_STAT_25K_009963", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.099, sens=0.94, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.277.", "meta": {"id": "WUA_STAT_25K_009964", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.3, μ=36.25, σ=17.65, compute z."}]}, "response": "z = 1.816.", "meta": {"id": "WUA_STAT_25K_009965", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.79, s=5.9, n=100, compute CI."}]}, "response": "95% CI ≈ [139.63, 141.95].", "meta": {"id": "WUA_STAT_25K_009966", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.24, compute E and Var."}]}, "response": "E[X]=2.88; Var[X]=2.189.", "meta": {"id": "WUA_STAT_25K_009967", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.35, compute E and Var."}]}, "response": "E[X]=2.80; Var[X]=1.820.", "meta": {"id": "WUA_STAT_25K_009968", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.66, s=30.05, n=146, compute CI."}]}, "response": "95% CI ≈ [130.79, 140.53].", "meta": {"id": "WUA_STAT_25K_009969", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.46, s=15.77, n=171, compute CI."}]}, "response": "95% CI ≈ [96.10, 100.82].", "meta": {"id": "WUA_STAT_25K_009970", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.82, s=1.89, n=198, compute CI."}]}, "response": "95% CI ≈ [92.56, 93.08].", "meta": {"id": "WUA_STAT_25K_009971", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.03, s=7.5, n=134, compute CI."}]}, "response": "95% CI ≈ [152.76, 155.30].", "meta": {"id": "WUA_STAT_25K_009972", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.07, compute E and Var."}]}, "response": "E[X]=2.59; Var[X]=2.409.", "meta": {"id": "WUA_STAT_25K_009973", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.49, compute E and Var."}]}, "response": "E[X]=12.25; Var[X]=6.248.", "meta": {"id": "WUA_STAT_25K_009974", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.62, compute E and Var."}]}, "response": "E[X]=19.84; Var[X]=7.539.", "meta": {"id": "WUA_STAT_25K_009975", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=165.41, s=1.29, n=123, compute CI."}]}, "response": "95% CI ≈ [165.18, 165.64].", "meta": {"id": "WUA_STAT_25K_009976", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.92, μ=15.52, σ=5.16, compute z."}]}, "response": "z = 0.465.", "meta": {"id": "WUA_STAT_25K_009977", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.15, μ=8.73, σ=13.45, compute z."}]}, "response": "z = 1.444.", "meta": {"id": "WUA_STAT_25K_009978", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.76, s=14.26, n=162, compute CI."}]}, "response": "95% CI ≈ [21.56, 25.96].", "meta": {"id": "WUA_STAT_25K_009979", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.84, μ=71.58, σ=21.08, compute z."}]}, "response": "z = -1.933.", "meta": {"id": "WUA_STAT_25K_009980", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.89, μ=57.7, σ=7.92, compute z."}]}, "response": "z = 0.403.", "meta": {"id": "WUA_STAT_25K_009981", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.85, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.345.", "meta": {"id": "WUA_STAT_25K_009982", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.39, s=29.11, n=104, compute CI."}]}, "response": "95% CI ≈ [57.80, 68.98].", "meta": {"id": "WUA_STAT_25K_009983", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.108, sens=0.89, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.915.", "meta": {"id": "WUA_STAT_25K_009984", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.47, compute E and Var."}]}, "response": "E[X]=15.51; Var[X]=8.220.", "meta": {"id": "WUA_STAT_25K_009985", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.62, s=27.59, n=65, compute CI."}]}, "response": "95% CI ≈ [106.91, 120.33].", "meta": {"id": "WUA_STAT_25K_009986", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.84, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.266.", "meta": {"id": "WUA_STAT_25K_009987", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.13, compute E and Var."}]}, "response": "E[X]=1.30; Var[X]=1.131.", "meta": {"id": "WUA_STAT_25K_009988", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.55, μ=49.21, σ=9.35, compute z."}]}, "response": "z = -0.605.", "meta": {"id": "WUA_STAT_25K_009989", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.77, compute E and Var."}]}, "response": "E[X]=8.47; Var[X]=1.948.", "meta": {"id": "WUA_STAT_25K_009990", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.95, s=21.66, n=29, compute CI."}]}, "response": "95% CI ≈ [129.07, 144.83].", "meta": {"id": "WUA_STAT_25K_009991", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.82, μ=29.56, σ=10.22, compute z."}]}, "response": "z = -2.812.", "meta": {"id": "WUA_STAT_25K_009992", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.94, μ=58.87, σ=14.46, compute z."}]}, "response": "z = -0.756.", "meta": {"id": "WUA_STAT_25K_009993", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.96, s=3.98, n=86, compute CI."}]}, "response": "95% CI ≈ [98.12, 99.80].", "meta": {"id": "WUA_STAT_25K_009994", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.09, sens=0.9, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.271.", "meta": {"id": "WUA_STAT_25K_009995", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.51, s=20.8, n=68, compute CI."}]}, "response": "95% CI ≈ [56.57, 66.45].", "meta": {"id": "WUA_STAT_25K_009996", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.047, sens=0.96, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.542.", "meta": {"id": "WUA_STAT_25K_009997", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.3, μ=74.77, σ=4.31, compute z."}]}, "response": "z = -12.174.", "meta": {"id": "WUA_STAT_25K_009998", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.94, μ=2.31, σ=2.38, compute z."}]}, "response": "z = 29.676.", "meta": {"id": "WUA_STAT_25K_009999", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.119, sens=0.96, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.342.", "meta": {"id": "WUA_STAT_25K_010000", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.13, compute E and Var."}]}, "response": "E[X]=2.73; Var[X]=2.375.", "meta": {"id": "WUA_STAT_25K_010001", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.86, s=9.65, n=147, compute CI."}]}, "response": "95% CI ≈ [23.30, 26.42].", "meta": {"id": "WUA_STAT_25K_010002", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.38, compute E and Var."}]}, "response": "E[X]=11.02; Var[X]=6.832.", "meta": {"id": "WUA_STAT_25K_010003", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.1, μ=95.57, σ=5.22, compute z."}]}, "response": "z = -7.944.", "meta": {"id": "WUA_STAT_25K_010004", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.21, compute E and Var."}]}, "response": "E[X]=4.83; Var[X]=3.816.", "meta": {"id": "WUA_STAT_25K_010005", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.32, μ=4.02, σ=15.12, compute z."}]}, "response": "z = 5.575.", "meta": {"id": "WUA_STAT_25K_010006", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.91, s=10.38, n=81, compute CI."}]}, "response": "95% CI ≈ [182.65, 187.17].", "meta": {"id": "WUA_STAT_25K_010007", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.79, s=17.97, n=38, compute CI."}]}, "response": "95% CI ≈ [185.08, 196.50].", "meta": {"id": "WUA_STAT_25K_010008", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.51, compute E and Var."}]}, "response": "E[X]=12.75; Var[X]=6.247.", "meta": {"id": "WUA_STAT_25K_010009", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.46, compute E and Var."}]}, "response": "E[X]=13.80; Var[X]=7.452.", "meta": {"id": "WUA_STAT_25K_010010", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.12, μ=53.15, σ=23.69, compute z."}]}, "response": "z = -0.550.", "meta": {"id": "WUA_STAT_25K_010011", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.096, sens=0.97, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.319.", "meta": {"id": "WUA_STAT_25K_010012", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.55, compute E and Var."}]}, "response": "E[X]=10.45; Var[X]=4.702.", "meta": {"id": "WUA_STAT_25K_010013", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.73, compute E and Var."}]}, "response": "E[X]=27.74; Var[X]=7.490.", "meta": {"id": "WUA_STAT_25K_010014", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.75, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.623.", "meta": {"id": "WUA_STAT_25K_010015", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.47, compute E and Var."}]}, "response": "E[X]=8.46; Var[X]=4.484.", "meta": {"id": "WUA_STAT_25K_010016", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.35, μ=41.14, σ=11.88, compute z."}]}, "response": "z = 3.974.", "meta": {"id": "WUA_STAT_25K_010017", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.047, sens=0.78, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.243.", "meta": {"id": "WUA_STAT_25K_010018", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.6, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_010019", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.2, s=17.16, n=51, compute CI."}]}, "response": "95% CI ≈ [112.49, 121.91].", "meta": {"id": "WUA_STAT_25K_010020", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=191.67, s=20.78, n=64, compute CI."}]}, "response": "95% CI ≈ [186.58, 196.76].", "meta": {"id": "WUA_STAT_25K_010021", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.1, s=14.04, n=198, compute CI."}]}, "response": "95% CI ≈ [130.14, 134.06].", "meta": {"id": "WUA_STAT_25K_010022", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.64, μ=89.07, σ=1.39, compute z."}]}, "response": "z = 0.410.", "meta": {"id": "WUA_STAT_25K_010023", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.118, sens=0.98, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.335.", "meta": {"id": "WUA_STAT_25K_010024", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.32, compute E and Var."}]}, "response": "E[X]=6.72; Var[X]=4.570.", "meta": {"id": "WUA_STAT_25K_010025", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.83, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.548.", "meta": {"id": "WUA_STAT_25K_010026", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.81, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.368.", "meta": {"id": "WUA_STAT_25K_010027", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.62, compute E and Var."}]}, "response": "E[X]=15.50; Var[X]=5.890.", "meta": {"id": "WUA_STAT_25K_010028", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.34, compute E and Var."}]}, "response": "E[X]=7.14; Var[X]=4.712.", "meta": {"id": "WUA_STAT_25K_010029", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.91, s=31.55, n=172, compute CI."}]}, "response": "95% CI ≈ [81.19, 90.63].", "meta": {"id": "WUA_STAT_25K_010030", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.82, s=8.08, n=123, compute CI."}]}, "response": "95% CI ≈ [136.39, 139.25].", "meta": {"id": "WUA_STAT_25K_010031", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.09, compute E and Var."}]}, "response": "E[X]=2.43; Var[X]=2.211.", "meta": {"id": "WUA_STAT_25K_010032", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.65, s=39.53, n=72, compute CI."}]}, "response": "95% CI ≈ [154.52, 172.78].", "meta": {"id": "WUA_STAT_25K_010033", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.4, μ=15.92, σ=20.61, compute z."}]}, "response": "z = -0.607.", "meta": {"id": "WUA_STAT_25K_010034", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.18, μ=45.91, σ=4.01, compute z."}]}, "response": "z = 10.042.", "meta": {"id": "WUA_STAT_25K_010035", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.84, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.922.", "meta": {"id": "WUA_STAT_25K_010036", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.78, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.375.", "meta": {"id": "WUA_STAT_25K_010037", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.99, s=39.43, n=63, compute CI."}]}, "response": "95% CI ≈ [12.25, 31.73].", "meta": {"id": "WUA_STAT_25K_010038", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.63, μ=78.35, σ=17.98, compute z."}]}, "response": "z = -0.541.", "meta": {"id": "WUA_STAT_25K_010039", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.59, μ=35.55, σ=22.66, compute z."}]}, "response": "z = 2.252.", "meta": {"id": "WUA_STAT_25K_010040", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.041, sens=0.97, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.242.", "meta": {"id": "WUA_STAT_25K_010041", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.1, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.890.", "meta": {"id": "WUA_STAT_25K_010042", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.64, μ=52.38, σ=5.66, compute z."}]}, "response": "z = -7.551.", "meta": {"id": "WUA_STAT_25K_010043", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.058, sens=0.94, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.325.", "meta": {"id": "WUA_STAT_25K_010044", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.28, s=24.08, n=162, compute CI."}]}, "response": "95% CI ≈ [86.57, 93.99].", "meta": {"id": "WUA_STAT_25K_010045", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.59, compute E and Var."}]}, "response": "E[X]=20.06; Var[X]=8.225.", "meta": {"id": "WUA_STAT_25K_010046", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.182, sens=0.98, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.509.", "meta": {"id": "WUA_STAT_25K_010047", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.37, compute E and Var."}]}, "response": "E[X]=5.55; Var[X]=3.496.", "meta": {"id": "WUA_STAT_25K_010048", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.1, s=23.41, n=179, compute CI."}]}, "response": "95% CI ≈ [58.67, 65.53].", "meta": {"id": "WUA_STAT_25K_010049", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.52, s=20.49, n=141, compute CI."}]}, "response": "95% CI ≈ [22.14, 28.90].", "meta": {"id": "WUA_STAT_25K_010050", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.63, compute E and Var."}]}, "response": "E[X]=17.01; Var[X]=6.294.", "meta": {"id": "WUA_STAT_25K_010051", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.72, compute E and Var."}]}, "response": "E[X]=25.20; Var[X]=7.056.", "meta": {"id": "WUA_STAT_25K_010052", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.28, compute E and Var."}]}, "response": "E[X]=4.48; Var[X]=3.226.", "meta": {"id": "WUA_STAT_25K_010053", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.169, sens=0.92, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.449.", "meta": {"id": "WUA_STAT_25K_010054", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.11, s=28.1, n=172, compute CI."}]}, "response": "95% CI ≈ [6.91, 15.31].", "meta": {"id": "WUA_STAT_25K_010055", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.6, compute E and Var."}]}, "response": "E[X]=23.40; Var[X]=9.360.", "meta": {"id": "WUA_STAT_25K_010056", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.92, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.357.", "meta": {"id": "WUA_STAT_25K_010057", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.43, compute E and Var."}]}, "response": "E[X]=3.01; Var[X]=1.716.", "meta": {"id": "WUA_STAT_25K_010058", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.89, s=9.95, n=110, compute CI."}]}, "response": "95% CI ≈ [184.03, 187.75].", "meta": {"id": "WUA_STAT_25K_010059", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.8, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.322.", "meta": {"id": "WUA_STAT_25K_010060", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.76, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.488.", "meta": {"id": "WUA_STAT_25K_010061", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.66, s=27.62, n=16, compute CI."}]}, "response": "95% CI ≈ [63.13, 90.19].", "meta": {"id": "WUA_STAT_25K_010062", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.105, sens=0.72, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.332.", "meta": {"id": "WUA_STAT_25K_010063", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.14, sens=0.9, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.463.", "meta": {"id": "WUA_STAT_25K_010064", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.75, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.898.", "meta": {"id": "WUA_STAT_25K_010065", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.52, s=13.28, n=74, compute CI."}]}, "response": "95% CI ≈ [183.49, 189.55].", "meta": {"id": "WUA_STAT_25K_010066", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.14, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=2.167.", "meta": {"id": "WUA_STAT_25K_010067", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.71, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.169.", "meta": {"id": "WUA_STAT_25K_010068", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.72, s=16.58, n=173, compute CI."}]}, "response": "95% CI ≈ [136.25, 141.19].", "meta": {"id": "WUA_STAT_25K_010069", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.63, s=15.9, n=148, compute CI."}]}, "response": "95% CI ≈ [118.07, 123.19].", "meta": {"id": "WUA_STAT_25K_010070", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.01, s=9.0, n=131, compute CI."}]}, "response": "95% CI ≈ [157.47, 160.55].", "meta": {"id": "WUA_STAT_25K_010071", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.44, compute E and Var."}]}, "response": "E[X]=15.84; Var[X]=8.870.", "meta": {"id": "WUA_STAT_25K_010072", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.2, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=4.480.", "meta": {"id": "WUA_STAT_25K_010073", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.106, sens=0.94, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.317.", "meta": {"id": "WUA_STAT_25K_010074", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.19, s=1.25, n=106, compute CI."}]}, "response": "95% CI ≈ [153.95, 154.43].", "meta": {"id": "WUA_STAT_25K_010075", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.74, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.122.", "meta": {"id": "WUA_STAT_25K_010076", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.91, s=34.38, n=70, compute CI."}]}, "response": "95% CI ≈ [124.86, 140.96].", "meta": {"id": "WUA_STAT_25K_010077", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.89, μ=81.83, σ=2.39, compute z."}]}, "response": "z = -30.100.", "meta": {"id": "WUA_STAT_25K_010078", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.51, compute E and Var."}]}, "response": "E[X]=11.22; Var[X]=5.498.", "meta": {"id": "WUA_STAT_25K_010079", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=19.26, s=24.04, n=74, compute CI."}]}, "response": "95% CI ≈ [13.78, 24.74].", "meta": {"id": "WUA_STAT_25K_010080", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.08, s=10.84, n=37, compute CI."}]}, "response": "95% CI ≈ [10.59, 17.57].", "meta": {"id": "WUA_STAT_25K_010081", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.89, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.622.", "meta": {"id": "WUA_STAT_25K_010082", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.41, μ=83.39, σ=18.2, compute z."}]}, "response": "z = -1.977.", "meta": {"id": "WUA_STAT_25K_010083", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.52, compute E and Var."}]}, "response": "E[X]=15.08; Var[X]=7.238.", "meta": {"id": "WUA_STAT_25K_010084", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.057, sens=0.88, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.181.", "meta": {"id": "WUA_STAT_25K_010085", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.79, μ=52.93, σ=24.32, compute z."}]}, "response": "z = 1.104.", "meta": {"id": "WUA_STAT_25K_010086", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.35, s=3.73, n=81, compute CI."}]}, "response": "95% CI ≈ [120.54, 122.16].", "meta": {"id": "WUA_STAT_25K_010087", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.62, s=6.14, n=41, compute CI."}]}, "response": "95% CI ≈ [34.74, 38.50].", "meta": {"id": "WUA_STAT_25K_010088", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.09, s=28.81, n=143, compute CI."}]}, "response": "95% CI ≈ [177.37, 186.81].", "meta": {"id": "WUA_STAT_25K_010089", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.06, μ=34.7, σ=9.94, compute z."}]}, "response": "z = -0.668.", "meta": {"id": "WUA_STAT_25K_010090", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.97, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.477.", "meta": {"id": "WUA_STAT_25K_010091", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.35, compute E and Var."}]}, "response": "E[X]=13.30; Var[X]=8.645.", "meta": {"id": "WUA_STAT_25K_010092", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.17, s=22.23, n=98, compute CI."}]}, "response": "95% CI ≈ [177.77, 186.57].", "meta": {"id": "WUA_STAT_25K_010093", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.27, μ=43.7, σ=20.85, compute z."}]}, "response": "z = -0.165.", "meta": {"id": "WUA_STAT_25K_010094", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.88, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.378.", "meta": {"id": "WUA_STAT_25K_010095", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.58, compute E and Var."}]}, "response": "E[X]=17.98; Var[X]=7.552.", "meta": {"id": "WUA_STAT_25K_010096", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.45, μ=25.17, σ=11.36, compute z."}]}, "response": "z = -1.560.", "meta": {"id": "WUA_STAT_25K_010097", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.71, μ=78.37, σ=18.68, compute z."}]}, "response": "z = -0.464.", "meta": {"id": "WUA_STAT_25K_010098", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.91, s=39.17, n=16, compute CI."}]}, "response": "95% CI ≈ [133.72, 172.10].", "meta": {"id": "WUA_STAT_25K_010099", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.85, μ=99.02, σ=4.3, compute z."}]}, "response": "z = -14.923.", "meta": {"id": "WUA_STAT_25K_010100", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.38, compute E and Var."}]}, "response": "E[X]=15.20; Var[X]=9.424.", "meta": {"id": "WUA_STAT_25K_010101", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.041, sens=0.75, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.391.", "meta": {"id": "WUA_STAT_25K_010102", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.14, compute E and Var."}]}, "response": "E[X]=4.62; Var[X]=3.973.", "meta": {"id": "WUA_STAT_25K_010103", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.01, s=5.37, n=194, compute CI."}]}, "response": "95% CI ≈ [42.25, 43.77].", "meta": {"id": "WUA_STAT_25K_010104", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.4, compute E and Var."}]}, "response": "E[X]=16.00; Var[X]=9.600.", "meta": {"id": "WUA_STAT_25K_010105", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.162, sens=0.77, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.415.", "meta": {"id": "WUA_STAT_25K_010106", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.91, μ=38.21, σ=3.66, compute z."}]}, "response": "z = 5.656.", "meta": {"id": "WUA_STAT_25K_010107", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.222, sens=0.77, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.733.", "meta": {"id": "WUA_STAT_25K_010108", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.24, μ=39.86, σ=16.48, compute z."}]}, "response": "z = 0.448.", "meta": {"id": "WUA_STAT_25K_010109", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.9, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.397.", "meta": {"id": "WUA_STAT_25K_010110", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.201, sens=0.72, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.516.", "meta": {"id": "WUA_STAT_25K_010111", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.24, μ=30.39, σ=13.39, compute z."}]}, "response": "z = 3.275.", "meta": {"id": "WUA_STAT_25K_010112", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.58, compute E and Var."}]}, "response": "E[X]=7.54; Var[X]=3.167.", "meta": {"id": "WUA_STAT_25K_010113", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.38, s=18.71, n=138, compute CI."}]}, "response": "95% CI ≈ [38.26, 44.50].", "meta": {"id": "WUA_STAT_25K_010114", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.034, sens=0.97, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.631.", "meta": {"id": "WUA_STAT_25K_010115", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.49, s=24.51, n=43, compute CI."}]}, "response": "95% CI ≈ [8.16, 22.82].", "meta": {"id": "WUA_STAT_25K_010116", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.31, μ=69.21, σ=12.17, compute z."}]}, "response": "z = -1.635.", "meta": {"id": "WUA_STAT_25K_010117", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.71, compute E and Var."}]}, "response": "E[X]=24.14; Var[X]=7.001.", "meta": {"id": "WUA_STAT_25K_010118", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.41, s=27.47, n=167, compute CI."}]}, "response": "95% CI ≈ [36.24, 44.58].", "meta": {"id": "WUA_STAT_25K_010119", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.25, sens=0.75, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.833.", "meta": {"id": "WUA_STAT_25K_010120", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.84, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.964.", "meta": {"id": "WUA_STAT_25K_010121", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.33, compute E and Var."}]}, "response": "E[X]=8.91; Var[X]=5.970.", "meta": {"id": "WUA_STAT_25K_010122", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.49, μ=2.27, σ=10.46, compute z."}]}, "response": "z = 9.008.", "meta": {"id": "WUA_STAT_25K_010123", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=96.4, s=17.12, n=40, compute CI."}]}, "response": "95% CI ≈ [91.09, 101.71].", "meta": {"id": "WUA_STAT_25K_010124", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.3, compute E and Var."}]}, "response": "E[X]=3.00; Var[X]=2.100.", "meta": {"id": "WUA_STAT_25K_010125", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.98, μ=23.39, σ=11.24, compute z."}]}, "response": "z = 4.323.", "meta": {"id": "WUA_STAT_25K_010126", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.28, compute E and Var."}]}, "response": "E[X]=5.32; Var[X]=3.830.", "meta": {"id": "WUA_STAT_25K_010127", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.37, compute E and Var."}]}, "response": "E[X]=10.36; Var[X]=6.527.", "meta": {"id": "WUA_STAT_25K_010128", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=148.5, s=29.81, n=53, compute CI."}]}, "response": "95% CI ≈ [140.47, 156.53].", "meta": {"id": "WUA_STAT_25K_010129", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.15, s=39.92, n=61, compute CI."}]}, "response": "95% CI ≈ [59.13, 79.17].", "meta": {"id": "WUA_STAT_25K_010130", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=124.29, s=38.23, n=81, compute CI."}]}, "response": "95% CI ≈ [115.96, 132.62].", "meta": {"id": "WUA_STAT_25K_010131", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.04, μ=39.44, σ=15.1, compute z."}]}, "response": "z = 2.623.", "meta": {"id": "WUA_STAT_25K_010132", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.38, compute E and Var."}]}, "response": "E[X]=14.06; Var[X]=8.717.", "meta": {"id": "WUA_STAT_25K_010133", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.057, sens=0.92, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.443.", "meta": {"id": "WUA_STAT_25K_010134", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=197.41, s=24.73, n=120, compute CI."}]}, "response": "95% CI ≈ [192.99, 201.83].", "meta": {"id": "WUA_STAT_25K_010135", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.37, compute E and Var."}]}, "response": "E[X]=10.36; Var[X]=6.527.", "meta": {"id": "WUA_STAT_25K_010136", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.62, μ=15.33, σ=17.37, compute z."}]}, "response": "z = 2.262.", "meta": {"id": "WUA_STAT_25K_010137", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.203, sens=0.91, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.502.", "meta": {"id": "WUA_STAT_25K_010138", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.55, s=9.1, n=56, compute CI."}]}, "response": "95% CI ≈ [170.17, 174.93].", "meta": {"id": "WUA_STAT_25K_010139", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=19.17, s=7.83, n=159, compute CI."}]}, "response": "95% CI ≈ [17.95, 20.39].", "meta": {"id": "WUA_STAT_25K_010140", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.65, compute E and Var."}]}, "response": "E[X]=25.35; Var[X]=8.873.", "meta": {"id": "WUA_STAT_25K_010141", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.65, μ=41.75, σ=4.31, compute z."}]}, "response": "z = -4.664.", "meta": {"id": "WUA_STAT_25K_010142", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.032, sens=0.77, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.104.", "meta": {"id": "WUA_STAT_25K_010143", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.42, s=39.46, n=168, compute CI."}]}, "response": "95% CI ≈ [23.45, 35.39].", "meta": {"id": "WUA_STAT_25K_010144", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.63, μ=61.26, σ=6.78, compute z."}]}, "response": "z = 3.299.", "meta": {"id": "WUA_STAT_25K_010145", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.69, s=11.62, n=71, compute CI."}]}, "response": "95% CI ≈ [136.99, 142.39].", "meta": {"id": "WUA_STAT_25K_010146", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=52.55, s=15.72, n=122, compute CI."}]}, "response": "95% CI ≈ [49.76, 55.34].", "meta": {"id": "WUA_STAT_25K_010147", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.041, sens=0.88, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.126.", "meta": {"id": "WUA_STAT_25K_010148", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.167, sens=0.89, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.598.", "meta": {"id": "WUA_STAT_25K_010149", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.221, sens=0.81, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.885.", "meta": {"id": "WUA_STAT_25K_010150", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.97, μ=39.05, σ=8.98, compute z."}]}, "response": "z = 4.223.", "meta": {"id": "WUA_STAT_25K_010151", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.26, compute E and Var."}]}, "response": "E[X]=6.24; Var[X]=4.618.", "meta": {"id": "WUA_STAT_25K_010152", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.78, μ=80.79, σ=12.77, compute z."}]}, "response": "z = -4.778.", "meta": {"id": "WUA_STAT_25K_010153", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.0, s=36.74, n=193, compute CI."}]}, "response": "95% CI ≈ [193.82, 204.18].", "meta": {"id": "WUA_STAT_25K_010154", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.25, sens=0.99, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.702.", "meta": {"id": "WUA_STAT_25K_010155", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.37, μ=72.58, σ=17.72, compute z."}]}, "response": "z = 0.948.", "meta": {"id": "WUA_STAT_25K_010156", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.24, sens=0.8, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.678.", "meta": {"id": "WUA_STAT_25K_010157", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.61, compute E and Var."}]}, "response": "E[X]=7.32; Var[X]=2.855.", "meta": {"id": "WUA_STAT_25K_010158", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.78, compute E and Var."}]}, "response": "E[X]=3.90; Var[X]=0.858.", "meta": {"id": "WUA_STAT_25K_010159", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.9, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.317.", "meta": {"id": "WUA_STAT_25K_010160", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.73, compute E and Var."}]}, "response": "E[X]=18.25; Var[X]=4.928.", "meta": {"id": "WUA_STAT_25K_010161", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.76, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.258.", "meta": {"id": "WUA_STAT_25K_010162", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.01, s=20.46, n=164, compute CI."}]}, "response": "95% CI ≈ [41.88, 48.14].", "meta": {"id": "WUA_STAT_25K_010163", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.83, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.529.", "meta": {"id": "WUA_STAT_25K_010164", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.52, μ=17.59, σ=2.65, compute z."}]}, "response": "z = 16.200.", "meta": {"id": "WUA_STAT_25K_010165", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.18, μ=8.92, σ=19.99, compute z."}]}, "response": "z = 2.364.", "meta": {"id": "WUA_STAT_25K_010166", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.62, s=24.88, n=186, compute CI."}]}, "response": "95% CI ≈ [135.04, 142.20].", "meta": {"id": "WUA_STAT_25K_010167", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.15, μ=54.24, σ=7.08, compute z."}]}, "response": "z = -1.849.", "meta": {"id": "WUA_STAT_25K_010168", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.24, s=12.14, n=125, compute CI."}]}, "response": "95% CI ≈ [171.11, 175.37].", "meta": {"id": "WUA_STAT_25K_010169", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.61, compute E and Var."}]}, "response": "E[X]=19.52; Var[X]=7.613.", "meta": {"id": "WUA_STAT_25K_010170", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.17, compute E and Var."}]}, "response": "E[X]=5.78; Var[X]=4.797.", "meta": {"id": "WUA_STAT_25K_010171", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.15, compute E and Var."}]}, "response": "E[X]=5.55; Var[X]=4.717.", "meta": {"id": "WUA_STAT_25K_010172", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.67, compute E and Var."}]}, "response": "E[X]=8.04; Var[X]=2.653.", "meta": {"id": "WUA_STAT_25K_010173", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=141.34, s=17.79, n=123, compute CI."}]}, "response": "95% CI ≈ [138.20, 144.48].", "meta": {"id": "WUA_STAT_25K_010174", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.046, sens=0.87, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.160.", "meta": {"id": "WUA_STAT_25K_010175", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.6, s=13.68, n=66, compute CI."}]}, "response": "95% CI ≈ [54.30, 60.90].", "meta": {"id": "WUA_STAT_25K_010176", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.5, compute E and Var."}]}, "response": "E[X]=6.50; Var[X]=3.250.", "meta": {"id": "WUA_STAT_25K_010177", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.244, sens=0.75, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.858.", "meta": {"id": "WUA_STAT_25K_010178", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.28, compute E and Var."}]}, "response": "E[X]=10.08; Var[X]=7.258.", "meta": {"id": "WUA_STAT_25K_010179", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=66.65, s=39.33, n=71, compute CI."}]}, "response": "95% CI ≈ [57.50, 75.80].", "meta": {"id": "WUA_STAT_25K_010180", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.58, μ=92.75, σ=1.03, compute z."}]}, "response": "z = -28.320.", "meta": {"id": "WUA_STAT_25K_010181", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.79, μ=48.27, σ=5.14, compute z."}]}, "response": "z = 5.938.", "meta": {"id": "WUA_STAT_25K_010182", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.148, sens=0.71, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.322.", "meta": {"id": "WUA_STAT_25K_010183", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.84, μ=19.29, σ=4.42, compute z."}]}, "response": "z = 8.043.", "meta": {"id": "WUA_STAT_25K_010184", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.54, compute E and Var."}]}, "response": "E[X]=18.36; Var[X]=8.446.", "meta": {"id": "WUA_STAT_25K_010185", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.76, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.444.", "meta": {"id": "WUA_STAT_25K_010186", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=104.49, s=34.31, n=86, compute CI."}]}, "response": "95% CI ≈ [97.24, 111.74].", "meta": {"id": "WUA_STAT_25K_010187", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=32.15, s=20.69, n=165, compute CI."}]}, "response": "95% CI ≈ [28.99, 35.31].", "meta": {"id": "WUA_STAT_25K_010188", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.167, sens=0.97, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.735.", "meta": {"id": "WUA_STAT_25K_010189", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.134, sens=0.97, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.714.", "meta": {"id": "WUA_STAT_25K_010190", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.15, μ=67.41, σ=15.74, compute z."}]}, "response": "z = -1.287.", "meta": {"id": "WUA_STAT_25K_010191", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.28, μ=35.97, σ=9.45, compute z."}]}, "response": "z = 1.197.", "meta": {"id": "WUA_STAT_25K_010192", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.87, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.250.", "meta": {"id": "WUA_STAT_25K_010193", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.51, s=18.34, n=121, compute CI."}]}, "response": "95% CI ≈ [80.24, 86.78].", "meta": {"id": "WUA_STAT_25K_010194", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.19, μ=22.43, σ=23.58, compute z."}]}, "response": "z = 3.170.", "meta": {"id": "WUA_STAT_25K_010195", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=195.87, s=2.96, n=107, compute CI."}]}, "response": "95% CI ≈ [195.31, 196.43].", "meta": {"id": "WUA_STAT_25K_010196", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.85, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.420.", "meta": {"id": "WUA_STAT_25K_010197", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.35, compute E and Var."}]}, "response": "E[X]=7.00; Var[X]=4.550.", "meta": {"id": "WUA_STAT_25K_010198", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.23, s=16.62, n=26, compute CI."}]}, "response": "95% CI ≈ [130.84, 143.62].", "meta": {"id": "WUA_STAT_25K_010199", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.69, s=10.76, n=193, compute CI."}]}, "response": "95% CI ≈ [131.17, 134.21].", "meta": {"id": "WUA_STAT_25K_010200", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.43, s=39.25, n=188, compute CI."}]}, "response": "95% CI ≈ [43.82, 55.04].", "meta": {"id": "WUA_STAT_25K_010201", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.247, sens=0.77, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.863.", "meta": {"id": "WUA_STAT_25K_010202", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.04, μ=38.04, σ=22.56, compute z."}]}, "response": "z = 2.394.", "meta": {"id": "WUA_STAT_25K_010203", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.1, μ=68.33, σ=17.57, compute z."}]}, "response": "z = -2.859.", "meta": {"id": "WUA_STAT_25K_010204", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.89, μ=96.96, σ=18.68, compute z."}]}, "response": "z = -3.002.", "meta": {"id": "WUA_STAT_25K_010205", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.134, sens=0.99, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.422.", "meta": {"id": "WUA_STAT_25K_010206", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.43, s=35.33, n=100, compute CI."}]}, "response": "95% CI ≈ [29.51, 43.35].", "meta": {"id": "WUA_STAT_25K_010207", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.05, μ=55.85, σ=6.18, compute z."}]}, "response": "z = -1.262.", "meta": {"id": "WUA_STAT_25K_010208", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.64, compute E and Var."}]}, "response": "E[X]=22.40; Var[X]=8.064.", "meta": {"id": "WUA_STAT_25K_010209", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.32, μ=39.21, σ=1.8, compute z."}]}, "response": "z = 10.617.", "meta": {"id": "WUA_STAT_25K_010210", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=197.46, s=21.25, n=191, compute CI."}]}, "response": "95% CI ≈ [194.45, 200.47].", "meta": {"id": "WUA_STAT_25K_010211", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.84, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.204.", "meta": {"id": "WUA_STAT_25K_010212", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.76, μ=68.71, σ=10.42, compute z."}]}, "response": "z = -4.410.", "meta": {"id": "WUA_STAT_25K_010213", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.52, μ=23.66, σ=15.27, compute z."}]}, "response": "z = 0.449.", "meta": {"id": "WUA_STAT_25K_010214", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.97, μ=61.71, σ=9.94, compute z."}]}, "response": "z = 0.831.", "meta": {"id": "WUA_STAT_25K_010215", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.51, compute E and Var."}]}, "response": "E[X]=13.26; Var[X]=6.497.", "meta": {"id": "WUA_STAT_25K_010216", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.13, μ=86.17, σ=5.44, compute z."}]}, "response": "z = -10.669.", "meta": {"id": "WUA_STAT_25K_010217", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.89, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.851.", "meta": {"id": "WUA_STAT_25K_010218", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=68.15, s=38.53, n=196, compute CI."}]}, "response": "95% CI ≈ [62.76, 73.54].", "meta": {"id": "WUA_STAT_25K_010219", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.0, s=13.48, n=168, compute CI."}]}, "response": "95% CI ≈ [175.96, 180.04].", "meta": {"id": "WUA_STAT_25K_010220", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.121, sens=0.79, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.731.", "meta": {"id": "WUA_STAT_25K_010221", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.93, μ=33.94, σ=23.06, compute z."}]}, "response": "z = 0.824.", "meta": {"id": "WUA_STAT_25K_010222", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.89, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.569.", "meta": {"id": "WUA_STAT_25K_010223", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.37, μ=20.96, σ=6.67, compute z."}]}, "response": "z = 1.561.", "meta": {"id": "WUA_STAT_25K_010224", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.9, s=24.92, n=47, compute CI."}]}, "response": "95% CI ≈ [132.78, 147.02].", "meta": {"id": "WUA_STAT_25K_010225", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.9, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.403.", "meta": {"id": "WUA_STAT_25K_010226", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=73.16, s=37.2, n=170, compute CI."}]}, "response": "95% CI ≈ [67.57, 78.75].", "meta": {"id": "WUA_STAT_25K_010227", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.21, compute E and Var."}]}, "response": "E[X]=7.77; Var[X]=6.138.", "meta": {"id": "WUA_STAT_25K_010228", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.16, μ=95.81, σ=12.21, compute z."}]}, "response": "z = -2.592.", "meta": {"id": "WUA_STAT_25K_010229", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.41, compute E and Var."}]}, "response": "E[X]=11.89; Var[X]=7.015.", "meta": {"id": "WUA_STAT_25K_010230", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.24, compute E and Var."}]}, "response": "E[X]=1.68; Var[X]=1.277.", "meta": {"id": "WUA_STAT_25K_010231", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.81, s=13.39, n=64, compute CI."}]}, "response": "95% CI ≈ [154.53, 161.09].", "meta": {"id": "WUA_STAT_25K_010232", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.69, μ=98.02, σ=5.22, compute z."}]}, "response": "z = -16.347.", "meta": {"id": "WUA_STAT_25K_010233", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.21, compute E and Var."}]}, "response": "E[X]=4.62; Var[X]=3.650.", "meta": {"id": "WUA_STAT_25K_010234", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.88, s=36.66, n=90, compute CI."}]}, "response": "95% CI ≈ [144.31, 159.45].", "meta": {"id": "WUA_STAT_25K_010235", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.225, sens=0.73, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.541.", "meta": {"id": "WUA_STAT_25K_010236", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.13, s=39.53, n=89, compute CI."}]}, "response": "95% CI ≈ [113.92, 130.34].", "meta": {"id": "WUA_STAT_25K_010237", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.05, compute E and Var."}]}, "response": "E[X]=0.85; Var[X]=0.807.", "meta": {"id": "WUA_STAT_25K_010238", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.61, compute E and Var."}]}, "response": "E[X]=20.13; Var[X]=7.851.", "meta": {"id": "WUA_STAT_25K_010239", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.96, s=18.24, n=130, compute CI."}]}, "response": "95% CI ≈ [131.82, 138.10].", "meta": {"id": "WUA_STAT_25K_010240", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.49, μ=15.57, σ=3.04, compute z."}]}, "response": "z = 11.816.", "meta": {"id": "WUA_STAT_25K_010241", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.96, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.421.", "meta": {"id": "WUA_STAT_25K_010242", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.85, s=38.44, n=170, compute CI."}]}, "response": "95% CI ≈ [158.07, 169.63].", "meta": {"id": "WUA_STAT_25K_010243", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.25, compute E and Var."}]}, "response": "E[X]=7.50; Var[X]=5.625.", "meta": {"id": "WUA_STAT_25K_010244", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=191.14, s=18.86, n=88, compute CI."}]}, "response": "95% CI ≈ [187.20, 195.08].", "meta": {"id": "WUA_STAT_25K_010245", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.81, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.286.", "meta": {"id": "WUA_STAT_25K_010246", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.51, compute E and Var."}]}, "response": "E[X]=12.75; Var[X]=6.247.", "meta": {"id": "WUA_STAT_25K_010247", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.137, sens=0.78, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.340.", "meta": {"id": "WUA_STAT_25K_010248", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.4, μ=18.74, σ=14.42, compute z."}]}, "response": "z = -0.093.", "meta": {"id": "WUA_STAT_25K_010249", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=56.64, s=28.57, n=199, compute CI."}]}, "response": "95% CI ≈ [52.67, 60.61].", "meta": {"id": "WUA_STAT_25K_010250", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.97, s=25.53, n=134, compute CI."}]}, "response": "95% CI ≈ [135.65, 144.29].", "meta": {"id": "WUA_STAT_25K_010251", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.12, μ=34.83, σ=10.0, compute z."}]}, "response": "z = 0.029.", "meta": {"id": "WUA_STAT_25K_010252", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.65, s=36.42, n=48, compute CI."}]}, "response": "95% CI ≈ [142.35, 162.95].", "meta": {"id": "WUA_STAT_25K_010253", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.74, μ=8.69, σ=16.55, compute z."}]}, "response": "z = 0.063.", "meta": {"id": "WUA_STAT_25K_010254", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.37, compute E and Var."}]}, "response": "E[X]=7.77; Var[X]=4.895.", "meta": {"id": "WUA_STAT_25K_010255", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.0, s=34.61, n=66, compute CI."}]}, "response": "95% CI ≈ [15.65, 32.35].", "meta": {"id": "WUA_STAT_25K_010256", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.175, sens=0.77, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.505.", "meta": {"id": "WUA_STAT_25K_010257", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.95, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.725.", "meta": {"id": "WUA_STAT_25K_010258", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.96, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.614.", "meta": {"id": "WUA_STAT_25K_010259", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.111, sens=0.97, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.302.", "meta": {"id": "WUA_STAT_25K_010260", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.74, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.479.", "meta": {"id": "WUA_STAT_25K_010261", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=64.85, s=25.88, n=185, compute CI."}]}, "response": "95% CI ≈ [61.12, 68.58].", "meta": {"id": "WUA_STAT_25K_010262", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.7, compute E and Var."}]}, "response": "E[X]=12.60; Var[X]=3.780.", "meta": {"id": "WUA_STAT_25K_010263", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.06, μ=67.09, σ=3.02, compute z."}]}, "response": "z = -15.904.", "meta": {"id": "WUA_STAT_25K_010264", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.2, s=1.32, n=154, compute CI."}]}, "response": "95% CI ≈ [79.99, 80.41].", "meta": {"id": "WUA_STAT_25K_010265", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.36, compute E and Var."}]}, "response": "E[X]=9.00; Var[X]=5.760.", "meta": {"id": "WUA_STAT_25K_010266", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.175, sens=0.77, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.368.", "meta": {"id": "WUA_STAT_25K_010267", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.29, compute E and Var."}]}, "response": "E[X]=6.09; Var[X]=4.324.", "meta": {"id": "WUA_STAT_25K_010268", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.28, compute E and Var."}]}, "response": "E[X]=4.48; Var[X]=3.226.", "meta": {"id": "WUA_STAT_25K_010269", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.91, μ=40.11, σ=22.89, compute z."}]}, "response": "z = 1.302.", "meta": {"id": "WUA_STAT_25K_010270", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.018, sens=0.94, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.117.", "meta": {"id": "WUA_STAT_25K_010271", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.58, μ=59.75, σ=11.65, compute z."}]}, "response": "z = -4.564.", "meta": {"id": "WUA_STAT_25K_010272", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.17, μ=20.63, σ=1.93, compute z."}]}, "response": "z = -2.311.", "meta": {"id": "WUA_STAT_25K_010273", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.93, μ=91.16, σ=14.01, compute z."}]}, "response": "z = -2.943.", "meta": {"id": "WUA_STAT_25K_010274", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.38, s=12.21, n=186, compute CI."}]}, "response": "95% CI ≈ [22.63, 26.13].", "meta": {"id": "WUA_STAT_25K_010275", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.43, μ=68.27, σ=8.56, compute z."}]}, "response": "z = -1.500.", "meta": {"id": "WUA_STAT_25K_010276", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.33, compute E and Var."}]}, "response": "E[X]=13.20; Var[X]=8.844.", "meta": {"id": "WUA_STAT_25K_010277", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.83, μ=13.64, σ=7.94, compute z."}]}, "response": "z = 8.840.", "meta": {"id": "WUA_STAT_25K_010278", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.65, μ=83.71, σ=2.22, compute z."}]}, "response": "z = -34.261.", "meta": {"id": "WUA_STAT_25K_010279", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.3, s=32.46, n=183, compute CI."}]}, "response": "95% CI ≈ [115.60, 125.00].", "meta": {"id": "WUA_STAT_25K_010280", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.72, compute E and Var."}]}, "response": "E[X]=5.76; Var[X]=1.613.", "meta": {"id": "WUA_STAT_25K_010281", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.35, compute E and Var."}]}, "response": "E[X]=8.75; Var[X]=5.688.", "meta": {"id": "WUA_STAT_25K_010282", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.5, compute E and Var."}]}, "response": "E[X]=19.00; Var[X]=9.500.", "meta": {"id": "WUA_STAT_25K_010283", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.8, compute E and Var."}]}, "response": "E[X]=25.60; Var[X]=5.120.", "meta": {"id": "WUA_STAT_25K_010284", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.8, μ=3.78, σ=11.03, compute z."}]}, "response": "z = 5.804.", "meta": {"id": "WUA_STAT_25K_010285", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.13, μ=82.3, σ=9.76, compute z."}]}, "response": "z = -4.013.", "meta": {"id": "WUA_STAT_25K_010286", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.71, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.222.", "meta": {"id": "WUA_STAT_25K_010287", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.41, s=32.22, n=13, compute CI."}]}, "response": "95% CI ≈ [176.90, 211.92].", "meta": {"id": "WUA_STAT_25K_010288", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.014, sens=0.85, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.060.", "meta": {"id": "WUA_STAT_25K_010289", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.41, μ=35.72, σ=17.15, compute z."}]}, "response": "z = -0.718.", "meta": {"id": "WUA_STAT_25K_010290", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.06, μ=11.72, σ=23.45, compute z."}]}, "response": "z = 2.616.", "meta": {"id": "WUA_STAT_25K_010291", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.39, compute E and Var."}]}, "response": "E[X]=9.36; Var[X]=5.710.", "meta": {"id": "WUA_STAT_25K_010292", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.69, μ=10.0, σ=15.86, compute z."}]}, "response": "z = 2.187.", "meta": {"id": "WUA_STAT_25K_010293", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.12, μ=43.35, σ=13.87, compute z."}]}, "response": "z = -2.756.", "meta": {"id": "WUA_STAT_25K_010294", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.89, μ=84.43, σ=13.22, compute z."}]}, "response": "z = -0.041.", "meta": {"id": "WUA_STAT_25K_010295", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.99, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.526.", "meta": {"id": "WUA_STAT_25K_010296", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.35, compute E and Var."}]}, "response": "E[X]=11.55; Var[X]=7.507.", "meta": {"id": "WUA_STAT_25K_010297", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.42, s=9.36, n=186, compute CI."}]}, "response": "95% CI ≈ [97.07, 99.77].", "meta": {"id": "WUA_STAT_25K_010298", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.16, compute E and Var."}]}, "response": "E[X]=4.80; Var[X]=4.032.", "meta": {"id": "WUA_STAT_25K_010299", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.77, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.470.", "meta": {"id": "WUA_STAT_25K_010300", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.36, compute E and Var."}]}, "response": "E[X]=3.24; Var[X]=2.074.", "meta": {"id": "WUA_STAT_25K_010301", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.24, compute E and Var."}]}, "response": "E[X]=4.56; Var[X]=3.466.", "meta": {"id": "WUA_STAT_25K_010302", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.92, μ=52.88, σ=14.11, compute z."}]}, "response": "z = 1.987.", "meta": {"id": "WUA_STAT_25K_010303", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.77, compute E and Var."}]}, "response": "E[X]=28.49; Var[X]=6.553.", "meta": {"id": "WUA_STAT_25K_010304", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.43, compute E and Var."}]}, "response": "E[X]=6.88; Var[X]=3.922.", "meta": {"id": "WUA_STAT_25K_010305", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.71, μ=98.73, σ=6.43, compute z."}]}, "response": "z = -3.580.", "meta": {"id": "WUA_STAT_25K_010306", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.5, μ=51.63, σ=3.11, compute z."}]}, "response": "z = 10.569.", "meta": {"id": "WUA_STAT_25K_010307", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.87, s=15.86, n=89, compute CI."}]}, "response": "95% CI ≈ [14.57, 21.17].", "meta": {"id": "WUA_STAT_25K_010308", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.43, compute E and Var."}]}, "response": "E[X]=10.32; Var[X]=5.882.", "meta": {"id": "WUA_STAT_25K_010309", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.39, compute E and Var."}]}, "response": "E[X]=7.41; Var[X]=4.520.", "meta": {"id": "WUA_STAT_25K_010310", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.12, compute E and Var."}]}, "response": "E[X]=2.16; Var[X]=1.901.", "meta": {"id": "WUA_STAT_25K_010311", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.69, compute E and Var."}]}, "response": "E[X]=4.14; Var[X]=1.283.", "meta": {"id": "WUA_STAT_25K_010312", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.06, compute E and Var."}]}, "response": "E[X]=0.48; Var[X]=0.451.", "meta": {"id": "WUA_STAT_25K_010313", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.09, s=3.13, n=41, compute CI."}]}, "response": "95% CI ≈ [33.13, 35.05].", "meta": {"id": "WUA_STAT_25K_010314", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.27, μ=49.66, σ=6.67, compute z."}]}, "response": "z = 3.540.", "meta": {"id": "WUA_STAT_25K_010315", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.3, s=11.49, n=118, compute CI."}]}, "response": "95% CI ≈ [77.23, 81.37].", "meta": {"id": "WUA_STAT_25K_010316", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.94, μ=92.94, σ=2.63, compute z."}]}, "response": "z = -3.422.", "meta": {"id": "WUA_STAT_25K_010317", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.7, s=26.79, n=193, compute CI."}]}, "response": "95% CI ≈ [114.92, 122.48].", "meta": {"id": "WUA_STAT_25K_010318", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.36, compute E and Var."}]}, "response": "E[X]=3.24; Var[X]=2.074.", "meta": {"id": "WUA_STAT_25K_010319", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.98, s=33.8, n=76, compute CI."}]}, "response": "95% CI ≈ [20.38, 35.58].", "meta": {"id": "WUA_STAT_25K_010320", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.04, μ=51.6, σ=1.43, compute z."}]}, "response": "z = 14.993.", "meta": {"id": "WUA_STAT_25K_010321", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.47, compute E and Var."}]}, "response": "E[X]=12.22; Var[X]=6.477.", "meta": {"id": "WUA_STAT_25K_010322", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.5, μ=76.02, σ=14.4, compute z."}]}, "response": "z = 1.561.", "meta": {"id": "WUA_STAT_25K_010323", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.2, s=33.71, n=164, compute CI."}]}, "response": "95% CI ≈ [171.04, 181.36].", "meta": {"id": "WUA_STAT_25K_010324", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.77, compute E and Var."}]}, "response": "E[X]=21.56; Var[X]=4.959.", "meta": {"id": "WUA_STAT_25K_010325", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.071, sens=0.91, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.199.", "meta": {"id": "WUA_STAT_25K_010326", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.35, μ=3.37, σ=15.44, compute z."}]}, "response": "z = 2.913.", "meta": {"id": "WUA_STAT_25K_010327", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.85, μ=22.71, σ=22.22, compute z."}]}, "response": "z = 0.906.", "meta": {"id": "WUA_STAT_25K_010328", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=103.56, s=21.97, n=90, compute CI."}]}, "response": "95% CI ≈ [99.02, 108.10].", "meta": {"id": "WUA_STAT_25K_010329", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.41, compute E and Var."}]}, "response": "E[X]=15.17; Var[X]=8.950.", "meta": {"id": "WUA_STAT_25K_010330", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.29, compute E and Var."}]}, "response": "E[X]=7.83; Var[X]=5.559.", "meta": {"id": "WUA_STAT_25K_010331", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.15, s=35.39, n=97, compute CI."}]}, "response": "95% CI ≈ [166.11, 180.19].", "meta": {"id": "WUA_STAT_25K_010332", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.241, sens=0.88, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.538.", "meta": {"id": "WUA_STAT_25K_010333", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.03, sens=0.72, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.110.", "meta": {"id": "WUA_STAT_25K_010334", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.09, compute E and Var."}]}, "response": "E[X]=0.81; Var[X]=0.737.", "meta": {"id": "WUA_STAT_25K_010335", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.3, μ=18.72, σ=18.27, compute z."}]}, "response": "z = 0.360.", "meta": {"id": "WUA_STAT_25K_010336", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.09, compute E and Var."}]}, "response": "E[X]=3.06; Var[X]=2.785.", "meta": {"id": "WUA_STAT_25K_010337", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.57, compute E and Var."}]}, "response": "E[X]=13.11; Var[X]=5.637.", "meta": {"id": "WUA_STAT_25K_010338", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.018, sens=0.87, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.117.", "meta": {"id": "WUA_STAT_25K_010339", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.51, μ=36.13, σ=23.42, compute z."}]}, "response": "z = -0.368.", "meta": {"id": "WUA_STAT_25K_010340", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.195, sens=0.97, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.566.", "meta": {"id": "WUA_STAT_25K_010341", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.65, compute E and Var."}]}, "response": "E[X]=4.55; Var[X]=1.592.", "meta": {"id": "WUA_STAT_25K_010342", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.26, s=3.45, n=120, compute CI."}]}, "response": "95% CI ≈ [129.64, 130.88].", "meta": {"id": "WUA_STAT_25K_010343", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.34, s=9.81, n=170, compute CI."}]}, "response": "95% CI ≈ [76.87, 79.81].", "meta": {"id": "WUA_STAT_25K_010344", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.53, compute E and Var."}]}, "response": "E[X]=13.78; Var[X]=6.477.", "meta": {"id": "WUA_STAT_25K_010345", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.3, compute E and Var."}]}, "response": "E[X]=9.90; Var[X]=6.930.", "meta": {"id": "WUA_STAT_25K_010346", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.74, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.325.", "meta": {"id": "WUA_STAT_25K_010347", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.85, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.855.", "meta": {"id": "WUA_STAT_25K_010348", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=116.33, s=12.09, n=20, compute CI."}]}, "response": "95% CI ≈ [111.03, 121.63].", "meta": {"id": "WUA_STAT_25K_010349", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.83, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.303.", "meta": {"id": "WUA_STAT_25K_010350", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.38, μ=75.09, σ=9.66, compute z."}]}, "response": "z = 0.134.", "meta": {"id": "WUA_STAT_25K_010351", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.249, sens=0.83, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.545.", "meta": {"id": "WUA_STAT_25K_010352", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.46, μ=60.23, σ=13.76, compute z."}]}, "response": "z = -1.364.", "meta": {"id": "WUA_STAT_25K_010353", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.59, s=3.94, n=90, compute CI."}]}, "response": "95% CI ≈ [186.78, 188.40].", "meta": {"id": "WUA_STAT_25K_010354", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.95, s=38.57, n=53, compute CI."}]}, "response": "95% CI ≈ [183.57, 204.33].", "meta": {"id": "WUA_STAT_25K_010355", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.77, μ=41.88, σ=16.08, compute z."}]}, "response": "z = -0.815.", "meta": {"id": "WUA_STAT_25K_010356", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.49, compute E and Var."}]}, "response": "E[X]=9.80; Var[X]=4.998.", "meta": {"id": "WUA_STAT_25K_010357", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.55, compute E and Var."}]}, "response": "E[X]=15.95; Var[X]=7.178.", "meta": {"id": "WUA_STAT_25K_010358", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.8, s=38.02, n=190, compute CI."}]}, "response": "95% CI ≈ [176.39, 187.21].", "meta": {"id": "WUA_STAT_25K_010359", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.64, μ=33.8, σ=15.76, compute z."}]}, "response": "z = 2.718.", "meta": {"id": "WUA_STAT_25K_010360", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=158.95, s=37.15, n=123, compute CI."}]}, "response": "95% CI ≈ [152.38, 165.52].", "meta": {"id": "WUA_STAT_25K_010361", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.7, compute E and Var."}]}, "response": "E[X]=9.80; Var[X]=2.940.", "meta": {"id": "WUA_STAT_25K_010362", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.3, μ=65.01, σ=3.4, compute z."}]}, "response": "z = -11.679.", "meta": {"id": "WUA_STAT_25K_010363", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.54, compute E and Var."}]}, "response": "E[X]=6.48; Var[X]=2.981.", "meta": {"id": "WUA_STAT_25K_010364", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.185, sens=0.73, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.380.", "meta": {"id": "WUA_STAT_25K_010365", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.1, compute E and Var."}]}, "response": "E[X]=1.80; Var[X]=1.620.", "meta": {"id": "WUA_STAT_25K_010366", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.76, compute E and Var."}]}, "response": "E[X]=14.44; Var[X]=3.466.", "meta": {"id": "WUA_STAT_25K_010367", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.85, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.555.", "meta": {"id": "WUA_STAT_25K_010368", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.55, compute E and Var."}]}, "response": "E[X]=18.70; Var[X]=8.415.", "meta": {"id": "WUA_STAT_25K_010369", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.4, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_010370", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.56, compute E and Var."}]}, "response": "E[X]=17.36; Var[X]=7.638.", "meta": {"id": "WUA_STAT_25K_010371", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.16, s=26.06, n=88, compute CI."}]}, "response": "95% CI ≈ [107.72, 118.60].", "meta": {"id": "WUA_STAT_25K_010372", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.3, s=36.37, n=108, compute CI."}]}, "response": "95% CI ≈ [80.44, 94.16].", "meta": {"id": "WUA_STAT_25K_010373", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.92, μ=10.81, σ=7.95, compute z."}]}, "response": "z = 6.932.", "meta": {"id": "WUA_STAT_25K_010374", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.24, compute E and Var."}]}, "response": "E[X]=3.84; Var[X]=2.918.", "meta": {"id": "WUA_STAT_25K_010375", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.024, sens=0.72, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.081.", "meta": {"id": "WUA_STAT_25K_010376", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.11, compute E and Var."}]}, "response": "E[X]=1.65; Var[X]=1.468.", "meta": {"id": "WUA_STAT_25K_010377", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.7, s=22.23, n=195, compute CI."}]}, "response": "95% CI ≈ [43.58, 49.82].", "meta": {"id": "WUA_STAT_25K_010378", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.03, sens=0.78, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.085.", "meta": {"id": "WUA_STAT_25K_010379", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.74, compute E and Var."}]}, "response": "E[X]=9.62; Var[X]=2.501.", "meta": {"id": "WUA_STAT_25K_010380", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.9, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.518.", "meta": {"id": "WUA_STAT_25K_010381", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.45, compute E and Var."}]}, "response": "E[X]=16.20; Var[X]=8.910.", "meta": {"id": "WUA_STAT_25K_010382", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.87, s=35.96, n=38, compute CI."}]}, "response": "95% CI ≈ [50.44, 73.30].", "meta": {"id": "WUA_STAT_25K_010383", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.87, s=38.98, n=157, compute CI."}]}, "response": "95% CI ≈ [130.77, 142.97].", "meta": {"id": "WUA_STAT_25K_010384", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.7, μ=72.51, σ=22.06, compute z."}]}, "response": "z = 0.371.", "meta": {"id": "WUA_STAT_25K_010385", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.17, μ=50.81, σ=10.35, compute z."}]}, "response": "z = -1.125.", "meta": {"id": "WUA_STAT_25K_010386", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.08, s=8.62, n=116, compute CI."}]}, "response": "95% CI ≈ [158.51, 161.65].", "meta": {"id": "WUA_STAT_25K_010387", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.25, μ=76.67, σ=15.34, compute z."}]}, "response": "z = -4.069.", "meta": {"id": "WUA_STAT_25K_010388", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.84, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.608.", "meta": {"id": "WUA_STAT_25K_010389", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.42, s=35.76, n=106, compute CI."}]}, "response": "95% CI ≈ [41.61, 55.23].", "meta": {"id": "WUA_STAT_25K_010390", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.46, s=34.38, n=170, compute CI."}]}, "response": "95% CI ≈ [56.29, 66.63].", "meta": {"id": "WUA_STAT_25K_010391", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.65, μ=41.86, σ=10.16, compute z."}]}, "response": "z = -0.021.", "meta": {"id": "WUA_STAT_25K_010392", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.25, compute E and Var."}]}, "response": "E[X]=2.75; Var[X]=2.062.", "meta": {"id": "WUA_STAT_25K_010393", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.12, compute E and Var."}]}, "response": "E[X]=2.64; Var[X]=2.323.", "meta": {"id": "WUA_STAT_25K_010394", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.38, compute E and Var."}]}, "response": "E[X]=11.02; Var[X]=6.832.", "meta": {"id": "WUA_STAT_25K_010395", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.76, μ=93.3, σ=21.77, compute z."}]}, "response": "z = -0.300.", "meta": {"id": "WUA_STAT_25K_010396", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.45, compute E and Var."}]}, "response": "E[X]=3.60; Var[X]=1.980.", "meta": {"id": "WUA_STAT_25K_010397", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.1, μ=45.19, σ=9.09, compute z."}]}, "response": "z = -0.780.", "meta": {"id": "WUA_STAT_25K_010398", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.66, compute E and Var."}]}, "response": "E[X]=15.18; Var[X]=5.161.", "meta": {"id": "WUA_STAT_25K_010399", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.91, s=35.02, n=93, compute CI."}]}, "response": "95% CI ≈ [80.79, 95.03].", "meta": {"id": "WUA_STAT_25K_010400", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.96, s=34.27, n=101, compute CI."}]}, "response": "95% CI ≈ [40.28, 53.64].", "meta": {"id": "WUA_STAT_25K_010401", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.98, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.512.", "meta": {"id": "WUA_STAT_25K_010402", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.48, s=26.65, n=184, compute CI."}]}, "response": "95% CI ≈ [146.63, 154.33].", "meta": {"id": "WUA_STAT_25K_010403", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.65, μ=57.09, σ=16.14, compute z."}]}, "response": "z = 2.637.", "meta": {"id": "WUA_STAT_25K_010404", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.9, μ=15.52, σ=16.15, compute z."}]}, "response": "z = 4.110.", "meta": {"id": "WUA_STAT_25K_010405", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.21, compute E and Var."}]}, "response": "E[X]=6.93; Var[X]=5.475.", "meta": {"id": "WUA_STAT_25K_010406", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.122, sens=0.7, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.251.", "meta": {"id": "WUA_STAT_25K_010407", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.57, compute E and Var."}]}, "response": "E[X]=2.85; Var[X]=1.226.", "meta": {"id": "WUA_STAT_25K_010408", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.2, compute E and Var."}]}, "response": "E[X]=5.00; Var[X]=4.000.", "meta": {"id": "WUA_STAT_25K_010409", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.09, s=39.16, n=156, compute CI."}]}, "response": "95% CI ≈ [71.94, 84.24].", "meta": {"id": "WUA_STAT_25K_010410", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.04, s=12.32, n=100, compute CI."}]}, "response": "95% CI ≈ [118.63, 123.45].", "meta": {"id": "WUA_STAT_25K_010411", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.61, compute E and Var."}]}, "response": "E[X]=16.47; Var[X]=6.423.", "meta": {"id": "WUA_STAT_25K_010412", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.25, s=31.45, n=148, compute CI."}]}, "response": "95% CI ≈ [181.18, 191.32].", "meta": {"id": "WUA_STAT_25K_010413", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.81, μ=84.58, σ=16.49, compute z."}]}, "response": "z = -2.472.", "meta": {"id": "WUA_STAT_25K_010414", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.95, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.294.", "meta": {"id": "WUA_STAT_25K_010415", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.32, s=23.89, n=27, compute CI."}]}, "response": "95% CI ≈ [145.31, 163.33].", "meta": {"id": "WUA_STAT_25K_010416", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.82, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.746.", "meta": {"id": "WUA_STAT_25K_010417", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.74, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.785.", "meta": {"id": "WUA_STAT_25K_010418", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.47, s=25.53, n=120, compute CI."}]}, "response": "95% CI ≈ [125.90, 135.04].", "meta": {"id": "WUA_STAT_25K_010419", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.55, compute E and Var."}]}, "response": "E[X]=13.20; Var[X]=5.940.", "meta": {"id": "WUA_STAT_25K_010420", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.78, s=9.17, n=95, compute CI."}]}, "response": "95% CI ≈ [63.94, 67.62].", "meta": {"id": "WUA_STAT_25K_010421", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.36, μ=44.13, σ=3.73, compute z."}]}, "response": "z = -9.322.", "meta": {"id": "WUA_STAT_25K_010422", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.02, s=31.41, n=121, compute CI."}]}, "response": "95% CI ≈ [140.42, 151.62].", "meta": {"id": "WUA_STAT_25K_010423", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.56, compute E and Var."}]}, "response": "E[X]=16.24; Var[X]=7.146.", "meta": {"id": "WUA_STAT_25K_010424", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.067, sens=0.7, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.148.", "meta": {"id": "WUA_STAT_25K_010425", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.51, compute E and Var."}]}, "response": "E[X]=16.83; Var[X]=8.247.", "meta": {"id": "WUA_STAT_25K_010426", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.26, compute E and Var."}]}, "response": "E[X]=8.58; Var[X]=6.349.", "meta": {"id": "WUA_STAT_25K_010427", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.15, s=31.5, n=139, compute CI."}]}, "response": "95% CI ≈ [112.91, 123.39].", "meta": {"id": "WUA_STAT_25K_010428", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=72.44, s=1.58, n=156, compute CI."}]}, "response": "95% CI ≈ [72.19, 72.69].", "meta": {"id": "WUA_STAT_25K_010429", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.15, μ=55.17, σ=7.43, compute z."}]}, "response": "z = -0.137.", "meta": {"id": "WUA_STAT_25K_010430", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.38, s=26.79, n=169, compute CI."}]}, "response": "95% CI ≈ [46.34, 54.42].", "meta": {"id": "WUA_STAT_25K_010431", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.09, μ=88.15, σ=5.34, compute z."}]}, "response": "z = -9.562.", "meta": {"id": "WUA_STAT_25K_010432", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.49, compute E and Var."}]}, "response": "E[X]=12.25; Var[X]=6.248.", "meta": {"id": "WUA_STAT_25K_010433", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.051, sens=0.85, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.395.", "meta": {"id": "WUA_STAT_25K_010434", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.23, s=24.12, n=117, compute CI."}]}, "response": "95% CI ≈ [22.86, 31.60].", "meta": {"id": "WUA_STAT_25K_010435", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.106, sens=0.88, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.777.", "meta": {"id": "WUA_STAT_25K_010436", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.61, s=31.42, n=61, compute CI."}]}, "response": "95% CI ≈ [181.73, 197.49].", "meta": {"id": "WUA_STAT_25K_010437", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.021, sens=0.88, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.076.", "meta": {"id": "WUA_STAT_25K_010438", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.96, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.729.", "meta": {"id": "WUA_STAT_25K_010439", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.22, μ=39.75, σ=11.42, compute z."}]}, "response": "z = 2.931.", "meta": {"id": "WUA_STAT_25K_010440", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.5, μ=12.45, σ=21.63, compute z."}]}, "response": "z = 1.436.", "meta": {"id": "WUA_STAT_25K_010441", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.179, sens=0.71, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.356.", "meta": {"id": "WUA_STAT_25K_010442", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.092, sens=0.92, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.757.", "meta": {"id": "WUA_STAT_25K_010443", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.72, μ=44.06, σ=15.83, compute z."}]}, "response": "z = 0.358.", "meta": {"id": "WUA_STAT_25K_010444", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.62, compute E and Var."}]}, "response": "E[X]=23.56; Var[X]=8.953.", "meta": {"id": "WUA_STAT_25K_010445", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=66.05, s=3.98, n=61, compute CI."}]}, "response": "95% CI ≈ [65.05, 67.05].", "meta": {"id": "WUA_STAT_25K_010446", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.72, s=11.41, n=198, compute CI."}]}, "response": "95% CI ≈ [37.13, 40.31].", "meta": {"id": "WUA_STAT_25K_010447", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=196.52, s=18.63, n=175, compute CI."}]}, "response": "95% CI ≈ [193.76, 199.28].", "meta": {"id": "WUA_STAT_25K_010448", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.68, compute E and Var."}]}, "response": "E[X]=26.52; Var[X]=8.486.", "meta": {"id": "WUA_STAT_25K_010449", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.19, s=25.46, n=70, compute CI."}]}, "response": "95% CI ≈ [136.23, 148.15].", "meta": {"id": "WUA_STAT_25K_010450", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.4, μ=90.66, σ=15.2, compute z."}]}, "response": "z = 0.443.", "meta": {"id": "WUA_STAT_25K_010451", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.36, μ=70.36, σ=17.36, compute z."}]}, "response": "z = -3.571.", "meta": {"id": "WUA_STAT_25K_010452", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.037, sens=0.88, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.104.", "meta": {"id": "WUA_STAT_25K_010453", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.76, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.459.", "meta": {"id": "WUA_STAT_25K_010454", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.26, compute E and Var."}]}, "response": "E[X]=9.88; Var[X]=7.311.", "meta": {"id": "WUA_STAT_25K_010455", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.63, compute E and Var."}]}, "response": "E[X]=18.90; Var[X]=6.993.", "meta": {"id": "WUA_STAT_25K_010456", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.96, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.327.", "meta": {"id": "WUA_STAT_25K_010457", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.68, s=23.24, n=45, compute CI."}]}, "response": "95% CI ≈ [91.89, 105.47].", "meta": {"id": "WUA_STAT_25K_010458", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.042, sens=0.99, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.165.", "meta": {"id": "WUA_STAT_25K_010459", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.38, compute E and Var."}]}, "response": "E[X]=14.44; Var[X]=8.953.", "meta": {"id": "WUA_STAT_25K_010460", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.48, s=25.92, n=56, compute CI."}]}, "response": "95% CI ≈ [137.69, 151.27].", "meta": {"id": "WUA_STAT_25K_010461", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.52, μ=50.19, σ=7.82, compute z."}]}, "response": "z = -0.341.", "meta": {"id": "WUA_STAT_25K_010462", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.81, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.406.", "meta": {"id": "WUA_STAT_25K_010463", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.7, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.744.", "meta": {"id": "WUA_STAT_25K_010464", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.78, μ=28.0, σ=17.76, compute z."}]}, "response": "z = -0.012.", "meta": {"id": "WUA_STAT_25K_010465", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.52, compute E and Var."}]}, "response": "E[X]=9.88; Var[X]=4.742.", "meta": {"id": "WUA_STAT_25K_010466", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.049, sens=0.86, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.240.", "meta": {"id": "WUA_STAT_25K_010467", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.84, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.278.", "meta": {"id": "WUA_STAT_25K_010468", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.84, μ=30.15, σ=21.82, compute z."}]}, "response": "z = -0.885.", "meta": {"id": "WUA_STAT_25K_010469", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.87, μ=48.49, σ=15.83, compute z."}]}, "response": "z = 0.150.", "meta": {"id": "WUA_STAT_25K_010470", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.88, s=8.21, n=17, compute CI."}]}, "response": "95% CI ≈ [31.98, 39.78].", "meta": {"id": "WUA_STAT_25K_010471", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.224, sens=0.97, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.549.", "meta": {"id": "WUA_STAT_25K_010472", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.85, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.486.", "meta": {"id": "WUA_STAT_25K_010473", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.41, compute E and Var."}]}, "response": "E[X]=8.20; Var[X]=4.838.", "meta": {"id": "WUA_STAT_25K_010474", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.08, μ=42.63, σ=18.4, compute z."}]}, "response": "z = 1.057.", "meta": {"id": "WUA_STAT_25K_010475", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.94, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.475.", "meta": {"id": "WUA_STAT_25K_010476", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.41, μ=13.77, σ=23.79, compute z."}]}, "response": "z = 2.843.", "meta": {"id": "WUA_STAT_25K_010477", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.94, μ=14.1, σ=12.49, compute z."}]}, "response": "z = 1.028.", "meta": {"id": "WUA_STAT_25K_010478", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.13, compute E and Var."}]}, "response": "E[X]=1.82; Var[X]=1.583.", "meta": {"id": "WUA_STAT_25K_010479", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.59, compute E and Var."}]}, "response": "E[X]=12.39; Var[X]=5.080.", "meta": {"id": "WUA_STAT_25K_010480", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=81.35, s=35.8, n=169, compute CI."}]}, "response": "95% CI ≈ [75.95, 86.75].", "meta": {"id": "WUA_STAT_25K_010481", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.0, s=10.83, n=191, compute CI."}]}, "response": "95% CI ≈ [173.46, 176.54].", "meta": {"id": "WUA_STAT_25K_010482", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.53, compute E and Var."}]}, "response": "E[X]=10.07; Var[X]=4.733.", "meta": {"id": "WUA_STAT_25K_010483", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.45, compute E and Var."}]}, "response": "E[X]=15.30; Var[X]=8.415.", "meta": {"id": "WUA_STAT_25K_010484", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.46, compute E and Var."}]}, "response": "E[X]=15.64; Var[X]=8.446.", "meta": {"id": "WUA_STAT_25K_010485", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.52, μ=42.49, σ=11.18, compute z."}]}, "response": "z = -2.323.", "meta": {"id": "WUA_STAT_25K_010486", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.53, s=14.95, n=164, compute CI."}]}, "response": "95% CI ≈ [140.24, 144.82].", "meta": {"id": "WUA_STAT_25K_010487", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.67, s=19.37, n=73, compute CI."}]}, "response": "95% CI ≈ [50.23, 59.11].", "meta": {"id": "WUA_STAT_25K_010488", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.77, s=11.69, n=35, compute CI."}]}, "response": "95% CI ≈ [30.90, 38.64].", "meta": {"id": "WUA_STAT_25K_010489", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.34, s=4.48, n=50, compute CI."}]}, "response": "95% CI ≈ [34.10, 36.58].", "meta": {"id": "WUA_STAT_25K_010490", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.64, μ=25.19, σ=9.57, compute z."}]}, "response": "z = -0.475.", "meta": {"id": "WUA_STAT_25K_010491", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.91, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.372.", "meta": {"id": "WUA_STAT_25K_010492", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.99, μ=84.31, σ=8.03, compute z."}]}, "response": "z = -8.508.", "meta": {"id": "WUA_STAT_25K_010493", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.92, μ=82.47, σ=20.5, compute z."}]}, "response": "z = -0.710.", "meta": {"id": "WUA_STAT_25K_010494", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.07, compute E and Var."}]}, "response": "E[X]=2.73; Var[X]=2.539.", "meta": {"id": "WUA_STAT_25K_010495", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.104, sens=0.73, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.514.", "meta": {"id": "WUA_STAT_25K_010496", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.83, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.626.", "meta": {"id": "WUA_STAT_25K_010497", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.31, s=11.76, n=161, compute CI."}]}, "response": "95% CI ≈ [184.49, 188.13].", "meta": {"id": "WUA_STAT_25K_010498", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.52, compute E and Var."}]}, "response": "E[X]=10.40; Var[X]=4.992.", "meta": {"id": "WUA_STAT_25K_010499", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.52, compute E and Var."}]}, "response": "E[X]=5.72; Var[X]=2.746.", "meta": {"id": "WUA_STAT_25K_010500", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.92, s=8.28, n=87, compute CI."}]}, "response": "95% CI ≈ [184.18, 187.66].", "meta": {"id": "WUA_STAT_25K_010501", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.83, μ=10.43, σ=7.28, compute z."}]}, "response": "z = 9.258.", "meta": {"id": "WUA_STAT_25K_010502", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.39, compute E and Var."}]}, "response": "E[X]=4.29; Var[X]=2.617.", "meta": {"id": "WUA_STAT_25K_010503", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.42, s=29.46, n=192, compute CI."}]}, "response": "95% CI ≈ [72.25, 80.59].", "meta": {"id": "WUA_STAT_25K_010504", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.25, s=37.68, n=35, compute CI."}]}, "response": "95% CI ≈ [85.77, 110.73].", "meta": {"id": "WUA_STAT_25K_010505", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.71, compute E and Var."}]}, "response": "E[X]=9.23; Var[X]=2.677.", "meta": {"id": "WUA_STAT_25K_010506", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.95, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.240.", "meta": {"id": "WUA_STAT_25K_010507", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.15, μ=75.83, σ=2.82, compute z."}]}, "response": "z = -25.064.", "meta": {"id": "WUA_STAT_25K_010508", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.25, s=27.54, n=116, compute CI."}]}, "response": "95% CI ≈ [92.24, 102.26].", "meta": {"id": "WUA_STAT_25K_010509", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.184, sens=0.99, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.817.", "meta": {"id": "WUA_STAT_25K_010510", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.18, compute E and Var."}]}, "response": "E[X]=4.86; Var[X]=3.985.", "meta": {"id": "WUA_STAT_25K_010511", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=183.06, s=38.52, n=28, compute CI."}]}, "response": "95% CI ≈ [168.79, 197.33].", "meta": {"id": "WUA_STAT_25K_010512", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.88, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.576.", "meta": {"id": "WUA_STAT_25K_010513", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.98, s=12.89, n=102, compute CI."}]}, "response": "95% CI ≈ [51.48, 56.48].", "meta": {"id": "WUA_STAT_25K_010514", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.02, μ=59.39, σ=16.73, compute z."}]}, "response": "z = -0.142.", "meta": {"id": "WUA_STAT_25K_010515", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.49, compute E and Var."}]}, "response": "E[X]=2.94; Var[X]=1.499.", "meta": {"id": "WUA_STAT_25K_010516", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.62, compute E and Var."}]}, "response": "E[X]=16.74; Var[X]=6.361.", "meta": {"id": "WUA_STAT_25K_010517", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.03, s=18.36, n=22, compute CI."}]}, "response": "95% CI ≈ [22.36, 37.70].", "meta": {"id": "WUA_STAT_25K_010518", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.29, μ=16.96, σ=20.64, compute z."}]}, "response": "z = 0.113.", "meta": {"id": "WUA_STAT_25K_010519", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.044, sens=0.85, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.151.", "meta": {"id": "WUA_STAT_25K_010520", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.58, s=13.98, n=85, compute CI."}]}, "response": "95% CI ≈ [82.61, 88.55].", "meta": {"id": "WUA_STAT_25K_010521", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.31, μ=19.07, σ=18.08, compute z."}]}, "response": "z = 3.387.", "meta": {"id": "WUA_STAT_25K_010522", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.15, μ=58.81, σ=1.99, compute z."}]}, "response": "z = -19.427.", "meta": {"id": "WUA_STAT_25K_010523", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.55, compute E and Var."}]}, "response": "E[X]=12.65; Var[X]=5.692.", "meta": {"id": "WUA_STAT_25K_010524", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.185, sens=0.83, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.758.", "meta": {"id": "WUA_STAT_25K_010525", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.36, compute E and Var."}]}, "response": "E[X]=11.88; Var[X]=7.603.", "meta": {"id": "WUA_STAT_25K_010526", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.37, μ=78.77, σ=23.93, compute z."}]}, "response": "z = 0.025.", "meta": {"id": "WUA_STAT_25K_010527", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.76, μ=40.64, σ=5.01, compute z."}]}, "response": "z = 1.022.", "meta": {"id": "WUA_STAT_25K_010528", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.98, μ=94.41, σ=19.04, compute z."}]}, "response": "z = -4.749.", "meta": {"id": "WUA_STAT_25K_010529", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.85, s=35.13, n=159, compute CI."}]}, "response": "95% CI ≈ [6.39, 17.31].", "meta": {"id": "WUA_STAT_25K_010530", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.62, compute E and Var."}]}, "response": "E[X]=11.16; Var[X]=4.241.", "meta": {"id": "WUA_STAT_25K_010531", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.72, compute E and Var."}]}, "response": "E[X]=16.56; Var[X]=4.637.", "meta": {"id": "WUA_STAT_25K_010532", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.66, μ=22.43, σ=14.81, compute z."}]}, "response": "z = 0.151.", "meta": {"id": "WUA_STAT_25K_010533", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.57, compute E and Var."}]}, "response": "E[X]=10.26; Var[X]=4.412.", "meta": {"id": "WUA_STAT_25K_010534", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.05, s=7.06, n=95, compute CI."}]}, "response": "95% CI ≈ [184.63, 187.47].", "meta": {"id": "WUA_STAT_25K_010535", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.13, compute E and Var."}]}, "response": "E[X]=2.47; Var[X]=2.149.", "meta": {"id": "WUA_STAT_25K_010536", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.74, μ=0.56, σ=17.88, compute z."}]}, "response": "z = 5.379.", "meta": {"id": "WUA_STAT_25K_010537", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.52, compute E and Var."}]}, "response": "E[X]=11.44; Var[X]=5.491.", "meta": {"id": "WUA_STAT_25K_010538", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.54, s=2.17, n=192, compute CI."}]}, "response": "95% CI ≈ [77.23, 77.85].", "meta": {"id": "WUA_STAT_25K_010539", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.38, μ=44.82, σ=6.09, compute z."}]}, "response": "z = -3.356.", "meta": {"id": "WUA_STAT_25K_010540", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.68, compute E and Var."}]}, "response": "E[X]=21.08; Var[X]=6.746.", "meta": {"id": "WUA_STAT_25K_010541", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.57, s=6.13, n=29, compute CI."}]}, "response": "95% CI ≈ [31.34, 35.80].", "meta": {"id": "WUA_STAT_25K_010542", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.78, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.366.", "meta": {"id": "WUA_STAT_25K_010543", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.173, sens=0.88, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.626.", "meta": {"id": "WUA_STAT_25K_010544", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.48, s=33.07, n=21, compute CI."}]}, "response": "95% CI ≈ [156.34, 184.62].", "meta": {"id": "WUA_STAT_25K_010545", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.39, compute E and Var."}]}, "response": "E[X]=12.09; Var[X]=7.375.", "meta": {"id": "WUA_STAT_25K_010546", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.4, μ=76.16, σ=5.23, compute z."}]}, "response": "z = -4.352.", "meta": {"id": "WUA_STAT_25K_010547", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.15, compute E and Var."}]}, "response": "E[X]=1.35; Var[X]=1.147.", "meta": {"id": "WUA_STAT_25K_010548", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.82, s=1.38, n=106, compute CI."}]}, "response": "95% CI ≈ [38.56, 39.08].", "meta": {"id": "WUA_STAT_25K_010549", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.14, sens=0.79, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.340.", "meta": {"id": "WUA_STAT_25K_010550", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.72, s=21.5, n=186, compute CI."}]}, "response": "95% CI ≈ [71.63, 77.81].", "meta": {"id": "WUA_STAT_25K_010551", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.2, compute E and Var."}]}, "response": "E[X]=7.80; Var[X]=6.240.", "meta": {"id": "WUA_STAT_25K_010552", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.13, μ=91.0, σ=23.08, compute z."}]}, "response": "z = -0.428.", "meta": {"id": "WUA_STAT_25K_010553", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.086, sens=0.75, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.391.", "meta": {"id": "WUA_STAT_25K_010554", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.47, s=32.54, n=131, compute CI."}]}, "response": "95% CI ≈ [139.90, 151.04].", "meta": {"id": "WUA_STAT_25K_010555", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.77, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.430.", "meta": {"id": "WUA_STAT_25K_010556", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.54, μ=90.04, σ=18.04, compute z."}]}, "response": "z = -2.245.", "meta": {"id": "WUA_STAT_25K_010557", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.87, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.550.", "meta": {"id": "WUA_STAT_25K_010558", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.98, s=24.85, n=184, compute CI."}]}, "response": "95% CI ≈ [195.39, 202.57].", "meta": {"id": "WUA_STAT_25K_010559", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.97, s=17.58, n=75, compute CI."}]}, "response": "95% CI ≈ [157.99, 165.95].", "meta": {"id": "WUA_STAT_25K_010560", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.42, s=29.91, n=55, compute CI."}]}, "response": "95% CI ≈ [149.52, 165.32].", "meta": {"id": "WUA_STAT_25K_010561", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.99, μ=72.06, σ=5.98, compute z."}]}, "response": "z = -10.212.", "meta": {"id": "WUA_STAT_25K_010562", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.74, compute E and Var."}]}, "response": "E[X]=12.58; Var[X]=3.271.", "meta": {"id": "WUA_STAT_25K_010563", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.24, compute E and Var."}]}, "response": "E[X]=6.48; Var[X]=4.925.", "meta": {"id": "WUA_STAT_25K_010564", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.82, μ=86.63, σ=6.04, compute z."}]}, "response": "z = -7.916.", "meta": {"id": "WUA_STAT_25K_010565", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.22, s=32.61, n=168, compute CI."}]}, "response": "95% CI ≈ [41.29, 51.15].", "meta": {"id": "WUA_STAT_25K_010566", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.089, sens=0.95, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.367.", "meta": {"id": "WUA_STAT_25K_010567", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.4, s=37.42, n=56, compute CI."}]}, "response": "95% CI ≈ [65.60, 85.20].", "meta": {"id": "WUA_STAT_25K_010568", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.54, μ=95.02, σ=13.76, compute z."}]}, "response": "z = -1.634.", "meta": {"id": "WUA_STAT_25K_010569", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.11, μ=36.16, σ=9.82, compute z."}]}, "response": "z = 5.087.", "meta": {"id": "WUA_STAT_25K_010570", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.82, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.203.", "meta": {"id": "WUA_STAT_25K_010571", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.75, μ=55.2, σ=18.55, compute z."}]}, "response": "z = -2.720.", "meta": {"id": "WUA_STAT_25K_010572", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.78, μ=32.39, σ=6.4, compute z."}]}, "response": "z = -0.564.", "meta": {"id": "WUA_STAT_25K_010573", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.11, s=16.48, n=83, compute CI."}]}, "response": "95% CI ≈ [51.56, 58.66].", "meta": {"id": "WUA_STAT_25K_010574", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=105.02, s=35.47, n=147, compute CI."}]}, "response": "95% CI ≈ [99.29, 110.75].", "meta": {"id": "WUA_STAT_25K_010575", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.72, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.040.", "meta": {"id": "WUA_STAT_25K_010576", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.79, compute E and Var."}]}, "response": "E[X]=20.54; Var[X]=4.313.", "meta": {"id": "WUA_STAT_25K_010577", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.53, μ=24.75, σ=21.17, compute z."}]}, "response": "z = 3.107.", "meta": {"id": "WUA_STAT_25K_010578", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.74, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.438.", "meta": {"id": "WUA_STAT_25K_010579", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.51, compute E and Var."}]}, "response": "E[X]=12.75; Var[X]=6.247.", "meta": {"id": "WUA_STAT_25K_010580", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.33, μ=38.0, σ=11.34, compute z."}]}, "response": "z = -2.352.", "meta": {"id": "WUA_STAT_25K_010581", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.56, μ=89.3, σ=14.23, compute z."}]}, "response": "z = -1.036.", "meta": {"id": "WUA_STAT_25K_010582", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.99, μ=45.54, σ=1.16, compute z."}]}, "response": "z = -22.888.", "meta": {"id": "WUA_STAT_25K_010583", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.77, s=18.52, n=176, compute CI."}]}, "response": "95% CI ≈ [187.03, 192.51].", "meta": {"id": "WUA_STAT_25K_010584", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.93, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.340.", "meta": {"id": "WUA_STAT_25K_010585", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.01, s=25.0, n=87, compute CI."}]}, "response": "95% CI ≈ [128.76, 139.26].", "meta": {"id": "WUA_STAT_25K_010586", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.62, compute E and Var."}]}, "response": "E[X]=23.56; Var[X]=8.953.", "meta": {"id": "WUA_STAT_25K_010587", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.95, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.776.", "meta": {"id": "WUA_STAT_25K_010588", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.32, compute E and Var."}]}, "response": "E[X]=11.20; Var[X]=7.616.", "meta": {"id": "WUA_STAT_25K_010589", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.156, sens=0.98, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.420.", "meta": {"id": "WUA_STAT_25K_010590", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.58, μ=86.49, σ=11.46, compute z."}]}, "response": "z = -2.959.", "meta": {"id": "WUA_STAT_25K_010591", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.68, μ=53.22, σ=22.44, compute z."}]}, "response": "z = 2.070.", "meta": {"id": "WUA_STAT_25K_010592", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.72, μ=94.73, σ=10.92, compute z."}]}, "response": "z = -2.382.", "meta": {"id": "WUA_STAT_25K_010593", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.7, compute E and Var."}]}, "response": "E[X]=19.60; Var[X]=5.880.", "meta": {"id": "WUA_STAT_25K_010594", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.46, μ=26.86, σ=15.4, compute z."}]}, "response": "z = 1.013.", "meta": {"id": "WUA_STAT_25K_010595", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.52, s=21.8, n=121, compute CI."}]}, "response": "95% CI ≈ [114.64, 122.40].", "meta": {"id": "WUA_STAT_25K_010596", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.042, sens=0.89, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.122.", "meta": {"id": "WUA_STAT_25K_010597", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.76, compute E and Var."}]}, "response": "E[X]=28.12; Var[X]=6.749.", "meta": {"id": "WUA_STAT_25K_010598", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.1, μ=73.29, σ=7.42, compute z."}]}, "response": "z = -1.239.", "meta": {"id": "WUA_STAT_25K_010599", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.07, μ=68.95, σ=1.76, compute z."}]}, "response": "z = 5.182.", "meta": {"id": "WUA_STAT_25K_010600", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.3, μ=15.61, σ=11.03, compute z."}]}, "response": "z = 2.057.", "meta": {"id": "WUA_STAT_25K_010601", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.246, sens=0.73, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.584.", "meta": {"id": "WUA_STAT_25K_010602", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.86, s=14.16, n=134, compute CI."}]}, "response": "95% CI ≈ [172.46, 177.26].", "meta": {"id": "WUA_STAT_25K_010603", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.08, compute E and Var."}]}, "response": "E[X]=2.80; Var[X]=2.576.", "meta": {"id": "WUA_STAT_25K_010604", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=93.72, s=5.05, n=45, compute CI."}]}, "response": "95% CI ≈ [92.24, 95.20].", "meta": {"id": "WUA_STAT_25K_010605", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.25, compute E and Var."}]}, "response": "E[X]=3.50; Var[X]=2.625.", "meta": {"id": "WUA_STAT_25K_010606", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=102.84, s=7.35, n=180, compute CI."}]}, "response": "95% CI ≈ [101.77, 103.91].", "meta": {"id": "WUA_STAT_25K_010607", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.76, compute E and Var."}]}, "response": "E[X]=26.60; Var[X]=6.384.", "meta": {"id": "WUA_STAT_25K_010608", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.058, sens=0.9, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.246.", "meta": {"id": "WUA_STAT_25K_010609", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.16, compute E and Var."}]}, "response": "E[X]=1.60; Var[X]=1.344.", "meta": {"id": "WUA_STAT_25K_010610", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.47, compute E and Var."}]}, "response": "E[X]=13.16; Var[X]=6.975.", "meta": {"id": "WUA_STAT_25K_010611", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.184, sens=0.92, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.522.", "meta": {"id": "WUA_STAT_25K_010612", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.6, s=13.92, n=167, compute CI."}]}, "response": "95% CI ≈ [186.49, 190.71].", "meta": {"id": "WUA_STAT_25K_010613", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=102.39, s=5.19, n=143, compute CI."}]}, "response": "95% CI ≈ [101.54, 103.24].", "meta": {"id": "WUA_STAT_25K_010614", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.21, s=39.13, n=106, compute CI."}]}, "response": "95% CI ≈ [17.76, 32.66].", "meta": {"id": "WUA_STAT_25K_010615", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.32, s=18.84, n=193, compute CI."}]}, "response": "95% CI ≈ [94.66, 99.98].", "meta": {"id": "WUA_STAT_25K_010616", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.97, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.587.", "meta": {"id": "WUA_STAT_25K_010617", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.6, μ=28.05, σ=18.79, compute z."}]}, "response": "z = -0.130.", "meta": {"id": "WUA_STAT_25K_010618", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.73, compute E and Var."}]}, "response": "E[X]=14.60; Var[X]=3.942.", "meta": {"id": "WUA_STAT_25K_010619", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.62, compute E and Var."}]}, "response": "E[X]=9.92; Var[X]=3.770.", "meta": {"id": "WUA_STAT_25K_010620", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.34, compute E and Var."}]}, "response": "E[X]=9.52; Var[X]=6.283.", "meta": {"id": "WUA_STAT_25K_010621", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.205, sens=0.86, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.597.", "meta": {"id": "WUA_STAT_25K_010622", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.238, sens=0.95, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.622.", "meta": {"id": "WUA_STAT_25K_010623", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.33, compute E and Var."}]}, "response": "E[X]=7.26; Var[X]=4.864.", "meta": {"id": "WUA_STAT_25K_010624", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.68, compute E and Var."}]}, "response": "E[X]=7.48; Var[X]=2.394.", "meta": {"id": "WUA_STAT_25K_010625", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.13, μ=98.12, σ=11.74, compute z."}]}, "response": "z = -1.021.", "meta": {"id": "WUA_STAT_25K_010626", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.21, μ=1.11, σ=7.51, compute z."}]}, "response": "z = 5.606.", "meta": {"id": "WUA_STAT_25K_010627", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.2, compute E and Var."}]}, "response": "E[X]=4.00; Var[X]=3.200.", "meta": {"id": "WUA_STAT_25K_010628", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.8, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.856.", "meta": {"id": "WUA_STAT_25K_010629", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.06, s=12.93, n=69, compute CI."}]}, "response": "95% CI ≈ [167.01, 173.11].", "meta": {"id": "WUA_STAT_25K_010630", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.56, compute E and Var."}]}, "response": "E[X]=16.80; Var[X]=7.392.", "meta": {"id": "WUA_STAT_25K_010631", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.6, compute E and Var."}]}, "response": "E[X]=7.80; Var[X]=3.120.", "meta": {"id": "WUA_STAT_25K_010632", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.52, μ=33.98, σ=8.39, compute z."}]}, "response": "z = 3.163.", "meta": {"id": "WUA_STAT_25K_010633", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.27, μ=49.61, σ=8.76, compute z."}]}, "response": "z = -1.295.", "meta": {"id": "WUA_STAT_25K_010634", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.042, sens=0.78, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.105.", "meta": {"id": "WUA_STAT_25K_010635", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.17, s=17.79, n=91, compute CI."}]}, "response": "95% CI ≈ [57.51, 64.83].", "meta": {"id": "WUA_STAT_25K_010636", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.097, sens=0.73, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.292.", "meta": {"id": "WUA_STAT_25K_010637", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.67, compute E and Var."}]}, "response": "E[X]=24.12; Var[X]=7.960.", "meta": {"id": "WUA_STAT_25K_010638", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.16, μ=53.02, σ=13.98, compute z."}]}, "response": "z = 1.727.", "meta": {"id": "WUA_STAT_25K_010639", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.27, compute E and Var."}]}, "response": "E[X]=7.56; Var[X]=5.519.", "meta": {"id": "WUA_STAT_25K_010640", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.055, sens=0.96, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.177.", "meta": {"id": "WUA_STAT_25K_010641", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.47, compute E and Var."}]}, "response": "E[X]=3.29; Var[X]=1.744.", "meta": {"id": "WUA_STAT_25K_010642", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.42, μ=17.49, σ=9.38, compute z."}]}, "response": "z = 5.110.", "meta": {"id": "WUA_STAT_25K_010643", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.17, sens=0.74, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.503.", "meta": {"id": "WUA_STAT_25K_010644", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.91, s=21.97, n=75, compute CI."}]}, "response": "95% CI ≈ [101.94, 111.88].", "meta": {"id": "WUA_STAT_25K_010645", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.06, compute E and Var."}]}, "response": "E[X]=1.56; Var[X]=1.466.", "meta": {"id": "WUA_STAT_25K_010646", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.41, compute E and Var."}]}, "response": "E[X]=14.35; Var[X]=8.467.", "meta": {"id": "WUA_STAT_25K_010647", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.011, sens=0.8, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.050.", "meta": {"id": "WUA_STAT_25K_010648", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.73, μ=20.43, σ=6.82, compute z."}]}, "response": "z = 5.176.", "meta": {"id": "WUA_STAT_25K_010649", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.92, μ=75.31, σ=4.2, compute z."}]}, "response": "z = -2.950.", "meta": {"id": "WUA_STAT_25K_010650", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.42, compute E and Var."}]}, "response": "E[X]=3.78; Var[X]=2.192.", "meta": {"id": "WUA_STAT_25K_010651", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.71, compute E and Var."}]}, "response": "E[X]=5.68; Var[X]=1.647.", "meta": {"id": "WUA_STAT_25K_010652", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.94, μ=15.87, σ=18.51, compute z."}]}, "response": "z = 1.030.", "meta": {"id": "WUA_STAT_25K_010653", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.83, μ=48.58, σ=1.39, compute z."}]}, "response": "z = -7.734.", "meta": {"id": "WUA_STAT_25K_010654", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.06, μ=25.33, σ=14.06, compute z."}]}, "response": "z = -1.584.", "meta": {"id": "WUA_STAT_25K_010655", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.72, compute E and Var."}]}, "response": "E[X]=27.36; Var[X]=7.661.", "meta": {"id": "WUA_STAT_25K_010656", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.233, sens=0.9, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.590.", "meta": {"id": "WUA_STAT_25K_010657", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.88, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.540.", "meta": {"id": "WUA_STAT_25K_010658", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.41, compute E and Var."}]}, "response": "E[X]=14.76; Var[X]=8.708.", "meta": {"id": "WUA_STAT_25K_010659", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.38, s=3.23, n=84, compute CI."}]}, "response": "95% CI ≈ [57.69, 59.07].", "meta": {"id": "WUA_STAT_25K_010660", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.066, sens=0.77, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.437.", "meta": {"id": "WUA_STAT_25K_010661", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.37, μ=14.84, σ=16.37, compute z."}]}, "response": "z = -0.151.", "meta": {"id": "WUA_STAT_25K_010662", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.55, s=13.88, n=197, compute CI."}]}, "response": "95% CI ≈ [188.61, 192.49].", "meta": {"id": "WUA_STAT_25K_010663", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.14, sens=0.83, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.342.", "meta": {"id": "WUA_STAT_25K_010664", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.1, μ=96.02, σ=11.71, compute z."}]}, "response": "z = -6.483.", "meta": {"id": "WUA_STAT_25K_010665", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.034, sens=0.97, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.363.", "meta": {"id": "WUA_STAT_25K_010666", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.88, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.429.", "meta": {"id": "WUA_STAT_25K_010667", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.41, compute E and Var."}]}, "response": "E[X]=3.69; Var[X]=2.177.", "meta": {"id": "WUA_STAT_25K_010668", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.0, μ=67.59, σ=12.14, compute z."}]}, "response": "z = -1.778.", "meta": {"id": "WUA_STAT_25K_010669", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.88, μ=45.01, σ=5.04, compute z."}]}, "response": "z = -3.597.", "meta": {"id": "WUA_STAT_25K_010670", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.96, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.426.", "meta": {"id": "WUA_STAT_25K_010671", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.46, compute E and Var."}]}, "response": "E[X]=6.44; Var[X]=3.478.", "meta": {"id": "WUA_STAT_25K_010672", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.71, μ=62.5, σ=8.78, compute z."}]}, "response": "z = -4.076.", "meta": {"id": "WUA_STAT_25K_010673", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.51, compute E and Var."}]}, "response": "E[X]=16.83; Var[X]=8.247.", "meta": {"id": "WUA_STAT_25K_010674", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.1, μ=6.09, σ=17.67, compute z."}]}, "response": "z = 4.528.", "meta": {"id": "WUA_STAT_25K_010675", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.79, compute E and Var."}]}, "response": "E[X]=23.70; Var[X]=4.977.", "meta": {"id": "WUA_STAT_25K_010676", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.028, sens=0.88, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.089.", "meta": {"id": "WUA_STAT_25K_010677", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.52, μ=61.71, σ=10.04, compute z."}]}, "response": "z = 1.375.", "meta": {"id": "WUA_STAT_25K_010678", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=51.65, s=18.18, n=85, compute CI."}]}, "response": "95% CI ≈ [47.79, 55.51].", "meta": {"id": "WUA_STAT_25K_010679", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.2, compute E and Var."}]}, "response": "E[X]=1.00; Var[X]=0.800.", "meta": {"id": "WUA_STAT_25K_010680", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.45, s=16.9, n=55, compute CI."}]}, "response": "95% CI ≈ [45.98, 54.92].", "meta": {"id": "WUA_STAT_25K_010681", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.248, sens=0.76, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.455.", "meta": {"id": "WUA_STAT_25K_010682", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.89, s=15.15, n=36, compute CI."}]}, "response": "95% CI ≈ [109.94, 119.84].", "meta": {"id": "WUA_STAT_25K_010683", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.79, compute E and Var."}]}, "response": "E[X]=14.22; Var[X]=2.986.", "meta": {"id": "WUA_STAT_25K_010684", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.217, sens=0.87, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.687.", "meta": {"id": "WUA_STAT_25K_010685", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.88, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.419.", "meta": {"id": "WUA_STAT_25K_010686", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.66, μ=43.25, σ=8.32, compute z."}]}, "response": "z = -2.595.", "meta": {"id": "WUA_STAT_25K_010687", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.4, compute E and Var."}]}, "response": "E[X]=15.60; Var[X]=9.360.", "meta": {"id": "WUA_STAT_25K_010688", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.39, compute E and Var."}]}, "response": "E[X]=3.51; Var[X]=2.141.", "meta": {"id": "WUA_STAT_25K_010689", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.176, sens=0.91, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.533.", "meta": {"id": "WUA_STAT_25K_010690", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.71, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.430.", "meta": {"id": "WUA_STAT_25K_010691", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.73, μ=86.71, σ=18.05, compute z."}]}, "response": "z = -3.877.", "meta": {"id": "WUA_STAT_25K_010692", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.69, s=12.72, n=132, compute CI."}]}, "response": "95% CI ≈ [43.52, 47.86].", "meta": {"id": "WUA_STAT_25K_010693", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.98, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.726.", "meta": {"id": "WUA_STAT_25K_010694", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.8, compute E and Var."}]}, "response": "E[X]=16.00; Var[X]=3.200.", "meta": {"id": "WUA_STAT_25K_010695", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.27, μ=14.63, σ=2.55, compute z."}]}, "response": "z = 31.624.", "meta": {"id": "WUA_STAT_25K_010696", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=72.28, s=36.1, n=121, compute CI."}]}, "response": "95% CI ≈ [65.85, 78.71].", "meta": {"id": "WUA_STAT_25K_010697", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.2, compute E and Var."}]}, "response": "E[X]=4.20; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_010698", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.56, compute E and Var."}]}, "response": "E[X]=6.16; Var[X]=2.710.", "meta": {"id": "WUA_STAT_25K_010699", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.56, compute E and Var."}]}, "response": "E[X]=11.76; Var[X]=5.174.", "meta": {"id": "WUA_STAT_25K_010700", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.73, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.793.", "meta": {"id": "WUA_STAT_25K_010701", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.36, compute E and Var."}]}, "response": "E[X]=2.16; Var[X]=1.382.", "meta": {"id": "WUA_STAT_25K_010702", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.04, μ=29.24, σ=23.4, compute z."}]}, "response": "z = -0.821.", "meta": {"id": "WUA_STAT_25K_010703", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.44, compute E and Var."}]}, "response": "E[X]=10.56; Var[X]=5.914.", "meta": {"id": "WUA_STAT_25K_010704", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.93, μ=64.34, σ=4.12, compute z."}]}, "response": "z = -1.070.", "meta": {"id": "WUA_STAT_25K_010705", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.98, s=12.39, n=17, compute CI."}]}, "response": "95% CI ≈ [115.09, 126.87].", "meta": {"id": "WUA_STAT_25K_010706", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.36, compute E and Var."}]}, "response": "E[X]=4.68; Var[X]=2.995.", "meta": {"id": "WUA_STAT_25K_010707", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.39, compute E and Var."}]}, "response": "E[X]=2.73; Var[X]=1.665.", "meta": {"id": "WUA_STAT_25K_010708", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.01, μ=95.49, σ=8.41, compute z."}]}, "response": "z = -6.359.", "meta": {"id": "WUA_STAT_25K_010709", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.32, s=2.42, n=115, compute CI."}]}, "response": "95% CI ≈ [74.88, 75.76].", "meta": {"id": "WUA_STAT_25K_010710", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.11, s=26.12, n=172, compute CI."}]}, "response": "95% CI ≈ [23.21, 31.01].", "meta": {"id": "WUA_STAT_25K_010711", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.49, compute E and Var."}]}, "response": "E[X]=13.72; Var[X]=6.997.", "meta": {"id": "WUA_STAT_25K_010712", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.65, compute E and Var."}]}, "response": "E[X]=18.20; Var[X]=6.370.", "meta": {"id": "WUA_STAT_25K_010713", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.86, μ=56.05, σ=19.96, compute z."}]}, "response": "z = -1.462.", "meta": {"id": "WUA_STAT_25K_010714", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.38, μ=21.05, σ=24.77, compute z."}]}, "response": "z = 0.901.", "meta": {"id": "WUA_STAT_25K_010715", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.16, compute E and Var."}]}, "response": "E[X]=5.28; Var[X]=4.435.", "meta": {"id": "WUA_STAT_25K_010716", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.99, s=39.42, n=168, compute CI."}]}, "response": "95% CI ≈ [137.03, 148.95].", "meta": {"id": "WUA_STAT_25K_010717", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.96, s=36.84, n=66, compute CI."}]}, "response": "95% CI ≈ [17.07, 34.85].", "meta": {"id": "WUA_STAT_25K_010718", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.62, s=11.31, n=49, compute CI."}]}, "response": "95% CI ≈ [75.45, 81.79].", "meta": {"id": "WUA_STAT_25K_010719", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.23, compute E and Var."}]}, "response": "E[X]=8.97; Var[X]=6.907.", "meta": {"id": "WUA_STAT_25K_010720", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.62, s=25.38, n=106, compute CI."}]}, "response": "95% CI ≈ [33.79, 43.45].", "meta": {"id": "WUA_STAT_25K_010721", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.75, compute E and Var."}]}, "response": "E[X]=27.75; Var[X]=6.938.", "meta": {"id": "WUA_STAT_25K_010722", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.17, sens=0.88, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.487.", "meta": {"id": "WUA_STAT_25K_010723", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.93, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.508.", "meta": {"id": "WUA_STAT_25K_010724", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.81, s=17.67, n=140, compute CI."}]}, "response": "95% CI ≈ [54.88, 60.74].", "meta": {"id": "WUA_STAT_25K_010725", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.86, μ=9.58, σ=9.47, compute z."}]}, "response": "z = 2.881.", "meta": {"id": "WUA_STAT_25K_010726", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.12, compute E and Var."}]}, "response": "E[X]=1.32; Var[X]=1.162.", "meta": {"id": "WUA_STAT_25K_010727", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.49, s=16.83, n=186, compute CI."}]}, "response": "95% CI ≈ [96.07, 100.91].", "meta": {"id": "WUA_STAT_25K_010728", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=183.08, s=31.72, n=73, compute CI."}]}, "response": "95% CI ≈ [175.80, 190.36].", "meta": {"id": "WUA_STAT_25K_010729", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.07, compute E and Var."}]}, "response": "E[X]=0.77; Var[X]=0.716.", "meta": {"id": "WUA_STAT_25K_010730", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.56, μ=0.7, σ=22.07, compute z."}]}, "response": "z = 3.392.", "meta": {"id": "WUA_STAT_25K_010731", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.01, s=17.42, n=120, compute CI."}]}, "response": "95% CI ≈ [116.89, 123.13].", "meta": {"id": "WUA_STAT_25K_010732", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.65, compute E and Var."}]}, "response": "E[X]=24.70; Var[X]=8.645.", "meta": {"id": "WUA_STAT_25K_010733", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.76, μ=37.72, σ=18.39, compute z."}]}, "response": "z = -1.955.", "meta": {"id": "WUA_STAT_25K_010734", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.48, compute E and Var."}]}, "response": "E[X]=2.40; Var[X]=1.248.", "meta": {"id": "WUA_STAT_25K_010735", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.83, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.842.", "meta": {"id": "WUA_STAT_25K_010736", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.68, s=25.11, n=94, compute CI."}]}, "response": "95% CI ≈ [49.60, 59.76].", "meta": {"id": "WUA_STAT_25K_010737", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.14, s=8.92, n=142, compute CI."}]}, "response": "95% CI ≈ [112.67, 115.61].", "meta": {"id": "WUA_STAT_25K_010738", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.81, s=39.99, n=127, compute CI."}]}, "response": "95% CI ≈ [166.85, 180.77].", "meta": {"id": "WUA_STAT_25K_010739", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=197.06, s=32.79, n=99, compute CI."}]}, "response": "95% CI ≈ [190.60, 203.52].", "meta": {"id": "WUA_STAT_25K_010740", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.032, sens=0.87, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.365.", "meta": {"id": "WUA_STAT_25K_010741", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.3, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.470.", "meta": {"id": "WUA_STAT_25K_010742", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.99, s=39.18, n=45, compute CI."}]}, "response": "95% CI ≈ [123.54, 146.44].", "meta": {"id": "WUA_STAT_25K_010743", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.09, compute E and Var."}]}, "response": "E[X]=3.24; Var[X]=2.948.", "meta": {"id": "WUA_STAT_25K_010744", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.187, sens=0.88, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.438.", "meta": {"id": "WUA_STAT_25K_010745", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.12, μ=6.31, σ=1.17, compute z."}]}, "response": "z = 50.265.", "meta": {"id": "WUA_STAT_25K_010746", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.13, μ=78.29, σ=23.22, compute z."}]}, "response": "z = 0.381.", "meta": {"id": "WUA_STAT_25K_010747", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.32, compute E and Var."}]}, "response": "E[X]=4.16; Var[X]=2.829.", "meta": {"id": "WUA_STAT_25K_010748", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.27, μ=45.32, σ=13.52, compute z."}]}, "response": "z = 1.180.", "meta": {"id": "WUA_STAT_25K_010749", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.85, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.186.", "meta": {"id": "WUA_STAT_25K_010750", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.187, sens=0.85, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.583.", "meta": {"id": "WUA_STAT_25K_010751", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.22, s=22.84, n=88, compute CI."}]}, "response": "95% CI ≈ [174.45, 183.99].", "meta": {"id": "WUA_STAT_25K_010752", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.55, μ=48.74, σ=22.52, compute z."}]}, "response": "z = 0.258.", "meta": {"id": "WUA_STAT_25K_010753", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.86, s=5.7, n=86, compute CI."}]}, "response": "95% CI ≈ [193.66, 196.06].", "meta": {"id": "WUA_STAT_25K_010754", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.67, compute E and Var."}]}, "response": "E[X]=12.73; Var[X]=4.201.", "meta": {"id": "WUA_STAT_25K_010755", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.56, compute E and Var."}]}, "response": "E[X]=21.28; Var[X]=9.363.", "meta": {"id": "WUA_STAT_25K_010756", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.21, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=1.991.", "meta": {"id": "WUA_STAT_25K_010757", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.41, s=18.34, n=10, compute CI."}]}, "response": "95% CI ≈ [3.04, 25.78].", "meta": {"id": "WUA_STAT_25K_010758", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.14, compute E and Var."}]}, "response": "E[X]=1.26; Var[X]=1.084.", "meta": {"id": "WUA_STAT_25K_010759", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.52, μ=93.18, σ=22.56, compute z."}]}, "response": "z = -0.650.", "meta": {"id": "WUA_STAT_25K_010760", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.018, sens=0.81, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.157.", "meta": {"id": "WUA_STAT_25K_010761", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.01, s=9.03, n=85, compute CI."}]}, "response": "95% CI ≈ [130.09, 133.93].", "meta": {"id": "WUA_STAT_25K_010762", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.143, sens=0.71, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.497.", "meta": {"id": "WUA_STAT_25K_010763", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=64.33, s=22.89, n=119, compute CI."}]}, "response": "95% CI ≈ [60.22, 68.44].", "meta": {"id": "WUA_STAT_25K_010764", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.29, μ=10.38, σ=17.49, compute z."}]}, "response": "z = 2.453.", "meta": {"id": "WUA_STAT_25K_010765", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.8, μ=79.86, σ=20.6, compute z."}]}, "response": "z = -2.236.", "meta": {"id": "WUA_STAT_25K_010766", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.47, compute E and Var."}]}, "response": "E[X]=16.92; Var[X]=8.968.", "meta": {"id": "WUA_STAT_25K_010767", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.47, s=36.42, n=23, compute CI."}]}, "response": "95% CI ≈ [-0.41, 29.35].", "meta": {"id": "WUA_STAT_25K_010768", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.45, compute E and Var."}]}, "response": "E[X]=7.65; Var[X]=4.208.", "meta": {"id": "WUA_STAT_25K_010769", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.38, s=32.63, n=13, compute CI."}]}, "response": "95% CI ≈ [143.64, 179.12].", "meta": {"id": "WUA_STAT_25K_010770", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.22, μ=57.95, σ=15.43, compute z."}]}, "response": "z = 0.017.", "meta": {"id": "WUA_STAT_25K_010771", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.54, compute E and Var."}]}, "response": "E[X]=17.28; Var[X]=7.949.", "meta": {"id": "WUA_STAT_25K_010772", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.122, sens=0.88, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.304.", "meta": {"id": "WUA_STAT_25K_010773", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.97, s=22.0, n=175, compute CI."}]}, "response": "95% CI ≈ [20.71, 27.23].", "meta": {"id": "WUA_STAT_25K_010774", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.66, compute E and Var."}]}, "response": "E[X]=25.08; Var[X]=8.527.", "meta": {"id": "WUA_STAT_25K_010775", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.55, s=30.02, n=152, compute CI."}]}, "response": "95% CI ≈ [22.78, 32.32].", "meta": {"id": "WUA_STAT_25K_010776", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.31, compute E and Var."}]}, "response": "E[X]=4.96; Var[X]=3.422.", "meta": {"id": "WUA_STAT_25K_010777", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.213, sens=0.86, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.593.", "meta": {"id": "WUA_STAT_25K_010778", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.5, μ=62.89, σ=13.16, compute z."}]}, "response": "z = 2.782.", "meta": {"id": "WUA_STAT_25K_010779", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.68, compute E and Var."}]}, "response": "E[X]=6.12; Var[X]=1.958.", "meta": {"id": "WUA_STAT_25K_010780", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.77, μ=85.97, σ=6.58, compute z."}]}, "response": "z = -10.365.", "meta": {"id": "WUA_STAT_25K_010781", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.86, s=29.21, n=19, compute CI."}]}, "response": "95% CI ≈ [160.73, 186.99].", "meta": {"id": "WUA_STAT_25K_010782", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.17, s=15.04, n=172, compute CI."}]}, "response": "95% CI ≈ [94.92, 99.42].", "meta": {"id": "WUA_STAT_25K_010783", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.18, μ=27.54, σ=4.2, compute z."}]}, "response": "z = 10.390.", "meta": {"id": "WUA_STAT_25K_010784", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.13, μ=79.12, σ=21.86, compute z."}]}, "response": "z = -2.561.", "meta": {"id": "WUA_STAT_25K_010785", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.68, μ=95.71, σ=20.3, compute z."}]}, "response": "z = -4.632.", "meta": {"id": "WUA_STAT_25K_010786", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.087, sens=0.91, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.250.", "meta": {"id": "WUA_STAT_25K_010787", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.13, μ=64.75, σ=18.93, compute z."}]}, "response": "z = -2.833.", "meta": {"id": "WUA_STAT_25K_010788", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.99, μ=79.0, σ=14.6, compute z."}]}, "response": "z = -3.699.", "meta": {"id": "WUA_STAT_25K_010789", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.17, compute E and Var."}]}, "response": "E[X]=6.63; Var[X]=5.503.", "meta": {"id": "WUA_STAT_25K_010790", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.247, sens=0.91, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.968.", "meta": {"id": "WUA_STAT_25K_010791", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.125, sens=0.86, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.754.", "meta": {"id": "WUA_STAT_25K_010792", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.29, compute E and Var."}]}, "response": "E[X]=10.44; Var[X]=7.412.", "meta": {"id": "WUA_STAT_25K_010793", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.46, compute E and Var."}]}, "response": "E[X]=5.06; Var[X]=2.732.", "meta": {"id": "WUA_STAT_25K_010794", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.69, μ=42.44, σ=17.78, compute z."}]}, "response": "z = 2.545.", "meta": {"id": "WUA_STAT_25K_010795", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=73.56, s=25.48, n=124, compute CI."}]}, "response": "95% CI ≈ [69.08, 78.04].", "meta": {"id": "WUA_STAT_25K_010796", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.33, s=12.51, n=73, compute CI."}]}, "response": "95% CI ≈ [142.46, 148.20].", "meta": {"id": "WUA_STAT_25K_010797", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.91, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.380.", "meta": {"id": "WUA_STAT_25K_010798", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.31, μ=83.16, σ=6.86, compute z."}]}, "response": "z = -4.643.", "meta": {"id": "WUA_STAT_25K_010799", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.77, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.367.", "meta": {"id": "WUA_STAT_25K_010800", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.45, compute E and Var."}]}, "response": "E[X]=8.55; Var[X]=4.703.", "meta": {"id": "WUA_STAT_25K_010801", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.3, s=34.26, n=175, compute CI."}]}, "response": "95% CI ≈ [112.22, 122.38].", "meta": {"id": "WUA_STAT_25K_010802", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.69, s=36.3, n=126, compute CI."}]}, "response": "95% CI ≈ [127.35, 140.03].", "meta": {"id": "WUA_STAT_25K_010803", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.45, μ=39.55, σ=9.3, compute z."}]}, "response": "z = 3.968.", "meta": {"id": "WUA_STAT_25K_010804", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.046, sens=0.82, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.189.", "meta": {"id": "WUA_STAT_25K_010805", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.24, μ=39.14, σ=18.05, compute z."}]}, "response": "z = 2.055.", "meta": {"id": "WUA_STAT_25K_010806", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.03, s=11.01, n=184, compute CI."}]}, "response": "95% CI ≈ [185.44, 188.62].", "meta": {"id": "WUA_STAT_25K_010807", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.201, sens=0.78, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.766.", "meta": {"id": "WUA_STAT_25K_010808", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.77, s=16.08, n=185, compute CI."}]}, "response": "95% CI ≈ [176.45, 181.09].", "meta": {"id": "WUA_STAT_25K_010809", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.66, compute E and Var."}]}, "response": "E[X]=16.50; Var[X]=5.610.", "meta": {"id": "WUA_STAT_25K_010810", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.58, compute E and Var."}]}, "response": "E[X]=4.06; Var[X]=1.705.", "meta": {"id": "WUA_STAT_25K_010811", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.14, μ=98.29, σ=19.01, compute z."}]}, "response": "z = -1.691.", "meta": {"id": "WUA_STAT_25K_010812", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.97, s=26.76, n=189, compute CI."}]}, "response": "95% CI ≈ [195.15, 202.79].", "meta": {"id": "WUA_STAT_25K_010813", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.65, compute E and Var."}]}, "response": "E[X]=23.40; Var[X]=8.190.", "meta": {"id": "WUA_STAT_25K_010814", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.027, sens=0.79, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.095.", "meta": {"id": "WUA_STAT_25K_010815", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.028, sens=0.99, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.120.", "meta": {"id": "WUA_STAT_25K_010816", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.86, μ=20.88, σ=13.53, compute z."}]}, "response": "z = 4.950.", "meta": {"id": "WUA_STAT_25K_010817", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=192.67, s=32.9, n=123, compute CI."}]}, "response": "95% CI ≈ [186.86, 198.48].", "meta": {"id": "WUA_STAT_25K_010818", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.84, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.946.", "meta": {"id": "WUA_STAT_25K_010819", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.25, compute E and Var."}]}, "response": "E[X]=7.75; Var[X]=5.812.", "meta": {"id": "WUA_STAT_25K_010820", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.2, sens=0.77, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.636.", "meta": {"id": "WUA_STAT_25K_010821", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.77, s=27.4, n=49, compute CI."}]}, "response": "95% CI ≈ [22.10, 37.44].", "meta": {"id": "WUA_STAT_25K_010822", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.071, sens=0.74, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.288.", "meta": {"id": "WUA_STAT_25K_010823", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=93.89, s=33.11, n=142, compute CI."}]}, "response": "95% CI ≈ [88.44, 99.34].", "meta": {"id": "WUA_STAT_25K_010824", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.44, μ=82.5, σ=7.09, compute z."}]}, "response": "z = 2.389.", "meta": {"id": "WUA_STAT_25K_010825", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.22, μ=96.97, σ=8.0, compute z."}]}, "response": "z = -1.969.", "meta": {"id": "WUA_STAT_25K_010826", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.47, compute E and Var."}]}, "response": "E[X]=9.40; Var[X]=4.982.", "meta": {"id": "WUA_STAT_25K_010827", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.74, compute E and Var."}]}, "response": "E[X]=28.12; Var[X]=7.311.", "meta": {"id": "WUA_STAT_25K_010828", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.17, s=37.63, n=188, compute CI."}]}, "response": "95% CI ≈ [172.79, 183.55].", "meta": {"id": "WUA_STAT_25K_010829", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.05, s=3.72, n=56, compute CI."}]}, "response": "95% CI ≈ [44.08, 46.02].", "meta": {"id": "WUA_STAT_25K_010830", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.197, sens=0.77, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.412.", "meta": {"id": "WUA_STAT_25K_010831", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=103.35, s=4.58, n=86, compute CI."}]}, "response": "95% CI ≈ [102.38, 104.32].", "meta": {"id": "WUA_STAT_25K_010832", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.21, s=36.08, n=77, compute CI."}]}, "response": "95% CI ≈ [36.15, 52.27].", "meta": {"id": "WUA_STAT_25K_010833", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.32, s=36.95, n=55, compute CI."}]}, "response": "95% CI ≈ [143.55, 163.09].", "meta": {"id": "WUA_STAT_25K_010834", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.92, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.451.", "meta": {"id": "WUA_STAT_25K_010835", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.24, s=11.05, n=185, compute CI."}]}, "response": "95% CI ≈ [155.65, 158.83].", "meta": {"id": "WUA_STAT_25K_010836", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.15, s=8.16, n=91, compute CI."}]}, "response": "95% CI ≈ [186.47, 189.83].", "meta": {"id": "WUA_STAT_25K_010837", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.73, compute E and Var."}]}, "response": "E[X]=18.25; Var[X]=4.928.", "meta": {"id": "WUA_STAT_25K_010838", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=28.41, s=19.64, n=166, compute CI."}]}, "response": "95% CI ≈ [25.42, 31.40].", "meta": {"id": "WUA_STAT_25K_010839", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.179, sens=0.87, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.422.", "meta": {"id": "WUA_STAT_25K_010840", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.2, s=24.85, n=188, compute CI."}]}, "response": "95% CI ≈ [131.65, 138.75].", "meta": {"id": "WUA_STAT_25K_010841", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.82, μ=1.49, σ=7.33, compute z."}]}, "response": "z = 4.547.", "meta": {"id": "WUA_STAT_25K_010842", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.15, s=7.27, n=87, compute CI."}]}, "response": "95% CI ≈ [109.62, 112.68].", "meta": {"id": "WUA_STAT_25K_010843", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.54, μ=28.38, σ=9.62, compute z."}]}, "response": "z = 6.150.", "meta": {"id": "WUA_STAT_25K_010844", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.39, μ=39.44, σ=16.83, compute z."}]}, "response": "z = 0.710.", "meta": {"id": "WUA_STAT_25K_010845", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.3, compute E and Var."}]}, "response": "E[X]=11.10; Var[X]=7.770.", "meta": {"id": "WUA_STAT_25K_010846", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.4, compute E and Var."}]}, "response": "E[X]=13.20; Var[X]=7.920.", "meta": {"id": "WUA_STAT_25K_010847", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.23, s=10.75, n=176, compute CI."}]}, "response": "95% CI ≈ [158.64, 161.82].", "meta": {"id": "WUA_STAT_25K_010848", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.58, μ=54.48, σ=18.95, compute z."}]}, "response": "z = 1.377.", "meta": {"id": "WUA_STAT_25K_010849", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.02, s=9.81, n=71, compute CI."}]}, "response": "95% CI ≈ [58.74, 63.30].", "meta": {"id": "WUA_STAT_25K_010850", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.224, sens=0.72, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.417.", "meta": {"id": "WUA_STAT_25K_010851", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.162, sens=0.84, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.765.", "meta": {"id": "WUA_STAT_25K_010852", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.47, s=21.38, n=128, compute CI."}]}, "response": "95% CI ≈ [174.77, 182.17].", "meta": {"id": "WUA_STAT_25K_010853", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.86, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.497.", "meta": {"id": "WUA_STAT_25K_010854", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.26, compute E and Var."}]}, "response": "E[X]=4.16; Var[X]=3.078.", "meta": {"id": "WUA_STAT_25K_010855", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.95, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.766.", "meta": {"id": "WUA_STAT_25K_010856", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.73, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.750.", "meta": {"id": "WUA_STAT_25K_010857", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.78, compute E and Var."}]}, "response": "E[X]=21.84; Var[X]=4.805.", "meta": {"id": "WUA_STAT_25K_010858", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.47, compute E and Var."}]}, "response": "E[X]=8.93; Var[X]=4.733.", "meta": {"id": "WUA_STAT_25K_010859", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.52, compute E and Var."}]}, "response": "E[X]=13.52; Var[X]=6.490.", "meta": {"id": "WUA_STAT_25K_010860", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=39.42, s=38.3, n=63, compute CI."}]}, "response": "95% CI ≈ [29.96, 48.88].", "meta": {"id": "WUA_STAT_25K_010861", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.07, μ=97.59, σ=6.64, compute z."}]}, "response": "z = -2.639.", "meta": {"id": "WUA_STAT_25K_010862", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.063, sens=0.87, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.203.", "meta": {"id": "WUA_STAT_25K_010863", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.76, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.248.", "meta": {"id": "WUA_STAT_25K_010864", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.31, compute E and Var."}]}, "response": "E[X]=10.85; Var[X]=7.486.", "meta": {"id": "WUA_STAT_25K_010865", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.67, compute E and Var."}]}, "response": "E[X]=6.70; Var[X]=2.211.", "meta": {"id": "WUA_STAT_25K_010866", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.78, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.269.", "meta": {"id": "WUA_STAT_25K_010867", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.42, s=11.79, n=187, compute CI."}]}, "response": "95% CI ≈ [185.73, 189.11].", "meta": {"id": "WUA_STAT_25K_010868", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.73, compute E and Var."}]}, "response": "E[X]=10.95; Var[X]=2.957.", "meta": {"id": "WUA_STAT_25K_010869", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.51, compute E and Var."}]}, "response": "E[X]=16.32; Var[X]=7.997.", "meta": {"id": "WUA_STAT_25K_010870", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.57, compute E and Var."}]}, "response": "E[X]=7.41; Var[X]=3.186.", "meta": {"id": "WUA_STAT_25K_010871", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.04, μ=87.79, σ=19.5, compute z."}]}, "response": "z = -3.269.", "meta": {"id": "WUA_STAT_25K_010872", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.12, μ=34.92, σ=19.88, compute z."}]}, "response": "z = -1.097.", "meta": {"id": "WUA_STAT_25K_010873", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.0, s=13.8, n=188, compute CI."}]}, "response": "95% CI ≈ [67.03, 70.97].", "meta": {"id": "WUA_STAT_25K_010874", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.45, compute E and Var."}]}, "response": "E[X]=16.20; Var[X]=8.910.", "meta": {"id": "WUA_STAT_25K_010875", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.62, μ=1.9, σ=17.21, compute z."}]}, "response": "z = 1.785.", "meta": {"id": "WUA_STAT_25K_010876", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.17, μ=16.04, σ=17.51, compute z."}]}, "response": "z = 3.891.", "meta": {"id": "WUA_STAT_25K_010877", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.23, compute E and Var."}]}, "response": "E[X]=1.38; Var[X]=1.063.", "meta": {"id": "WUA_STAT_25K_010878", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.93, μ=90.1, σ=12.55, compute z."}]}, "response": "z = -0.253.", "meta": {"id": "WUA_STAT_25K_010879", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.77, μ=33.22, σ=17.23, compute z."}]}, "response": "z = 3.572.", "meta": {"id": "WUA_STAT_25K_010880", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=100.82, s=1.64, n=51, compute CI."}]}, "response": "95% CI ≈ [100.37, 101.27].", "meta": {"id": "WUA_STAT_25K_010881", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.74, compute E and Var."}]}, "response": "E[X]=9.62; Var[X]=2.501.", "meta": {"id": "WUA_STAT_25K_010882", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.45, μ=99.96, σ=17.32, compute z."}]}, "response": "z = -5.226.", "meta": {"id": "WUA_STAT_25K_010883", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.83, μ=98.17, σ=8.07, compute z."}]}, "response": "z = -0.042.", "meta": {"id": "WUA_STAT_25K_010884", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.218, sens=0.83, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.623.", "meta": {"id": "WUA_STAT_25K_010885", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.237, sens=0.85, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.868.", "meta": {"id": "WUA_STAT_25K_010886", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.21, μ=32.64, σ=22.23, compute z."}]}, "response": "z = 0.296.", "meta": {"id": "WUA_STAT_25K_010887", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.034, sens=0.73, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.079.", "meta": {"id": "WUA_STAT_25K_010888", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.31, compute E and Var."}]}, "response": "E[X]=10.54; Var[X]=7.273.", "meta": {"id": "WUA_STAT_25K_010889", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.53, compute E and Var."}]}, "response": "E[X]=5.83; Var[X]=2.740.", "meta": {"id": "WUA_STAT_25K_010890", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.03, μ=28.01, σ=11.4, compute z."}]}, "response": "z = 5.265.", "meta": {"id": "WUA_STAT_25K_010891", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.0, s=11.48, n=185, compute CI."}]}, "response": "95% CI ≈ [10.35, 13.65].", "meta": {"id": "WUA_STAT_25K_010892", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.43, compute E and Var."}]}, "response": "E[X]=12.90; Var[X]=7.353.", "meta": {"id": "WUA_STAT_25K_010893", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.165, sens=0.78, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.372.", "meta": {"id": "WUA_STAT_25K_010894", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.91, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.230.", "meta": {"id": "WUA_STAT_25K_010895", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.05, s=13.34, n=170, compute CI."}]}, "response": "95% CI ≈ [116.04, 120.06].", "meta": {"id": "WUA_STAT_25K_010896", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.46, μ=71.87, σ=1.12, compute z."}]}, "response": "z = 4.991.", "meta": {"id": "WUA_STAT_25K_010897", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.62, compute E and Var."}]}, "response": "E[X]=19.22; Var[X]=7.304.", "meta": {"id": "WUA_STAT_25K_010898", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.34, compute E and Var."}]}, "response": "E[X]=13.26; Var[X]=8.752.", "meta": {"id": "WUA_STAT_25K_010899", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.41, compute E and Var."}]}, "response": "E[X]=8.61; Var[X]=5.080.", "meta": {"id": "WUA_STAT_25K_010900", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.94, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.191.", "meta": {"id": "WUA_STAT_25K_010901", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.31, μ=11.3, σ=7.02, compute z."}]}, "response": "z = 6.981.", "meta": {"id": "WUA_STAT_25K_010902", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.71, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.264.", "meta": {"id": "WUA_STAT_25K_010903", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.39, μ=30.78, σ=5.88, compute z."}]}, "response": "z = 3.165.", "meta": {"id": "WUA_STAT_25K_010904", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.81, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.379.", "meta": {"id": "WUA_STAT_25K_010905", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.32, μ=84.94, σ=4.57, compute z."}]}, "response": "z = -15.891.", "meta": {"id": "WUA_STAT_25K_010906", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.93, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.197.", "meta": {"id": "WUA_STAT_25K_010907", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.09, sens=0.91, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.563.", "meta": {"id": "WUA_STAT_25K_010908", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.67, s=32.37, n=113, compute CI."}]}, "response": "95% CI ≈ [52.70, 64.64].", "meta": {"id": "WUA_STAT_25K_010909", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.25, μ=46.58, σ=6.61, compute z."}]}, "response": "z = -5.648.", "meta": {"id": "WUA_STAT_25K_010910", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=168.76, s=14.2, n=112, compute CI."}]}, "response": "95% CI ≈ [166.13, 171.39].", "meta": {"id": "WUA_STAT_25K_010911", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.41, compute E and Var."}]}, "response": "E[X]=10.25; Var[X]=6.048.", "meta": {"id": "WUA_STAT_25K_010912", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.7, compute E and Var."}]}, "response": "E[X]=23.80; Var[X]=7.140.", "meta": {"id": "WUA_STAT_25K_010913", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.16, s=4.28, n=133, compute CI."}]}, "response": "95% CI ≈ [79.43, 80.89].", "meta": {"id": "WUA_STAT_25K_010914", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.7, s=15.03, n=25, compute CI."}]}, "response": "95% CI ≈ [100.81, 112.59].", "meta": {"id": "WUA_STAT_25K_010915", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.054, sens=0.79, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.429.", "meta": {"id": "WUA_STAT_25K_010916", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.6, compute E and Var."}]}, "response": "E[X]=22.20; Var[X]=8.880.", "meta": {"id": "WUA_STAT_25K_010917", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.78, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.332.", "meta": {"id": "WUA_STAT_25K_010918", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.031, sens=0.93, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.110.", "meta": {"id": "WUA_STAT_25K_010919", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.13, μ=58.52, σ=8.42, compute z."}]}, "response": "z = -1.709.", "meta": {"id": "WUA_STAT_25K_010920", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.06, μ=91.21, σ=17.67, compute z."}]}, "response": "z = -1.027.", "meta": {"id": "WUA_STAT_25K_010921", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.72, μ=66.59, σ=9.29, compute z."}]}, "response": "z = -0.632.", "meta": {"id": "WUA_STAT_25K_010922", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.9, μ=70.9, σ=11.89, compute z."}]}, "response": "z = 1.682.", "meta": {"id": "WUA_STAT_25K_010923", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.76, μ=57.53, σ=8.22, compute z."}]}, "response": "z = -5.811.", "meta": {"id": "WUA_STAT_25K_010924", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.42, μ=47.9, σ=15.17, compute z."}]}, "response": "z = -2.603.", "meta": {"id": "WUA_STAT_25K_010925", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.79, compute E and Var."}]}, "response": "E[X]=26.07; Var[X]=5.475.", "meta": {"id": "WUA_STAT_25K_010926", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=105.23, s=5.89, n=146, compute CI."}]}, "response": "95% CI ≈ [104.27, 106.19].", "meta": {"id": "WUA_STAT_25K_010927", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.02, s=38.67, n=28, compute CI."}]}, "response": "95% CI ≈ [172.70, 201.34].", "meta": {"id": "WUA_STAT_25K_010928", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.93, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.724.", "meta": {"id": "WUA_STAT_25K_010929", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.46, μ=51.72, σ=10.09, compute z."}]}, "response": "z = 4.533.", "meta": {"id": "WUA_STAT_25K_010930", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.35, μ=55.08, σ=24.61, compute z."}]}, "response": "z = -1.289.", "meta": {"id": "WUA_STAT_25K_010931", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.026, sens=0.9, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.077.", "meta": {"id": "WUA_STAT_25K_010932", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.3, s=12.3, n=122, compute CI."}]}, "response": "95% CI ≈ [68.12, 72.48].", "meta": {"id": "WUA_STAT_25K_010933", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.27, compute E and Var."}]}, "response": "E[X]=6.48; Var[X]=4.730.", "meta": {"id": "WUA_STAT_25K_010934", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=42.35, s=10.32, n=41, compute CI."}]}, "response": "95% CI ≈ [39.19, 45.51].", "meta": {"id": "WUA_STAT_25K_010935", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.175, sens=0.88, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.675.", "meta": {"id": "WUA_STAT_25K_010936", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.99, s=27.34, n=94, compute CI."}]}, "response": "95% CI ≈ [71.46, 82.52].", "meta": {"id": "WUA_STAT_25K_010937", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.214, sens=0.79, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.843.", "meta": {"id": "WUA_STAT_25K_010938", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.99, μ=2.96, σ=16.85, compute z."}]}, "response": "z = 3.325.", "meta": {"id": "WUA_STAT_25K_010939", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.72, μ=3.97, σ=11.35, compute z."}]}, "response": "z = 8.348.", "meta": {"id": "WUA_STAT_25K_010940", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.072, sens=0.73, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.168.", "meta": {"id": "WUA_STAT_25K_010941", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.29, compute E and Var."}]}, "response": "E[X]=3.48; Var[X]=2.471.", "meta": {"id": "WUA_STAT_25K_010942", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.27, compute E and Var."}]}, "response": "E[X]=4.59; Var[X]=3.351.", "meta": {"id": "WUA_STAT_25K_010943", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.78, compute E and Var."}]}, "response": "E[X]=29.64; Var[X]=6.521.", "meta": {"id": "WUA_STAT_25K_010944", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.11, μ=24.65, σ=11.28, compute z."}]}, "response": "z = 3.941.", "meta": {"id": "WUA_STAT_25K_010945", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.42, s=27.83, n=88, compute CI."}]}, "response": "95% CI ≈ [93.61, 105.23].", "meta": {"id": "WUA_STAT_25K_010946", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.38, compute E and Var."}]}, "response": "E[X]=8.36; Var[X]=5.183.", "meta": {"id": "WUA_STAT_25K_010947", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=164.94, s=2.82, n=26, compute CI."}]}, "response": "95% CI ≈ [163.86, 166.02].", "meta": {"id": "WUA_STAT_25K_010948", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.157, sens=0.71, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.346.", "meta": {"id": "WUA_STAT_25K_010949", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.78, s=20.05, n=188, compute CI."}]}, "response": "95% CI ≈ [122.91, 128.65].", "meta": {"id": "WUA_STAT_25K_010950", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.218, sens=0.82, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.604.", "meta": {"id": "WUA_STAT_25K_010951", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.29, compute E and Var."}]}, "response": "E[X]=1.45; Var[X]=1.029.", "meta": {"id": "WUA_STAT_25K_010952", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.1, s=23.83, n=84, compute CI."}]}, "response": "95% CI ≈ [148.00, 158.20].", "meta": {"id": "WUA_STAT_25K_010953", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.32, compute E and Var."}]}, "response": "E[X]=11.20; Var[X]=7.616.", "meta": {"id": "WUA_STAT_25K_010954", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.71, μ=85.67, σ=18.25, compute z."}]}, "response": "z = -3.559.", "meta": {"id": "WUA_STAT_25K_010955", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.37, compute E and Var."}]}, "response": "E[X]=3.70; Var[X]=2.331.", "meta": {"id": "WUA_STAT_25K_010956", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=51.6, s=25.73, n=10, compute CI."}]}, "response": "95% CI ≈ [35.65, 67.55].", "meta": {"id": "WUA_STAT_25K_010957", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.49, μ=16.28, σ=18.24, compute z."}]}, "response": "z = 1.766.", "meta": {"id": "WUA_STAT_25K_010958", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.34, compute E and Var."}]}, "response": "E[X]=13.26; Var[X]=8.752.", "meta": {"id": "WUA_STAT_25K_010959", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.97, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.221.", "meta": {"id": "WUA_STAT_25K_010960", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.063, sens=0.76, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.719.", "meta": {"id": "WUA_STAT_25K_010961", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=39.82, s=5.19, n=93, compute CI."}]}, "response": "95% CI ≈ [38.77, 40.87].", "meta": {"id": "WUA_STAT_25K_010962", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.7, s=4.32, n=145, compute CI."}]}, "response": "95% CI ≈ [123.00, 124.40].", "meta": {"id": "WUA_STAT_25K_010963", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.49, compute E and Var."}]}, "response": "E[X]=6.86; Var[X]=3.499.", "meta": {"id": "WUA_STAT_25K_010964", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.249, sens=0.94, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.621.", "meta": {"id": "WUA_STAT_25K_010965", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.2, compute E and Var."}]}, "response": "E[X]=7.80; Var[X]=6.240.", "meta": {"id": "WUA_STAT_25K_010966", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.13, compute E and Var."}]}, "response": "E[X]=4.16; Var[X]=3.619.", "meta": {"id": "WUA_STAT_25K_010967", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.227, sens=0.95, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.538.", "meta": {"id": "WUA_STAT_25K_010968", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.08, μ=79.07, σ=19.69, compute z."}]}, "response": "z = 0.762.", "meta": {"id": "WUA_STAT_25K_010969", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.3, compute E and Var."}]}, "response": "E[X]=6.60; Var[X]=4.620.", "meta": {"id": "WUA_STAT_25K_010970", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.85, μ=21.03, σ=14.06, compute z."}]}, "response": "z = 5.606.", "meta": {"id": "WUA_STAT_25K_010971", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.82, s=16.45, n=71, compute CI."}]}, "response": "95% CI ≈ [84.99, 92.65].", "meta": {"id": "WUA_STAT_25K_010972", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.89, μ=10.64, σ=13.19, compute z."}]}, "response": "z = 1.763.", "meta": {"id": "WUA_STAT_25K_010973", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.24, μ=66.13, σ=18.03, compute z."}]}, "response": "z = -2.212.", "meta": {"id": "WUA_STAT_25K_010974", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.71, μ=60.07, σ=8.52, compute z."}]}, "response": "z = 0.545.", "meta": {"id": "WUA_STAT_25K_010975", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.36, compute E and Var."}]}, "response": "E[X]=9.00; Var[X]=5.760.", "meta": {"id": "WUA_STAT_25K_010976", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.09, compute E and Var."}]}, "response": "E[X]=1.17; Var[X]=1.065.", "meta": {"id": "WUA_STAT_25K_010977", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.062, sens=0.8, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.325.", "meta": {"id": "WUA_STAT_25K_010978", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.18, μ=93.83, σ=14.38, compute z."}]}, "response": "z = 0.372.", "meta": {"id": "WUA_STAT_25K_010979", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=72.52, s=26.43, n=125, compute CI."}]}, "response": "95% CI ≈ [67.89, 77.15].", "meta": {"id": "WUA_STAT_25K_010980", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.18, sens=0.78, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.631.", "meta": {"id": "WUA_STAT_25K_010981", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.221, sens=0.79, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.651.", "meta": {"id": "WUA_STAT_25K_010982", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.41, μ=33.38, σ=14.13, compute z."}]}, "response": "z = 2.550.", "meta": {"id": "WUA_STAT_25K_010983", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.66, s=3.0, n=105, compute CI."}]}, "response": "95% CI ≈ [85.09, 86.23].", "meta": {"id": "WUA_STAT_25K_010984", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.52, s=13.24, n=124, compute CI."}]}, "response": "95% CI ≈ [115.19, 119.85].", "meta": {"id": "WUA_STAT_25K_010985", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.56, compute E and Var."}]}, "response": "E[X]=15.12; Var[X]=6.653.", "meta": {"id": "WUA_STAT_25K_010986", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.57, s=8.44, n=164, compute CI."}]}, "response": "95% CI ≈ [75.28, 77.86].", "meta": {"id": "WUA_STAT_25K_010987", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.33, compute E and Var."}]}, "response": "E[X]=8.25; Var[X]=5.527.", "meta": {"id": "WUA_STAT_25K_010988", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.28, s=14.9, n=122, compute CI."}]}, "response": "95% CI ≈ [176.64, 181.92].", "meta": {"id": "WUA_STAT_25K_010989", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.63, s=18.89, n=178, compute CI."}]}, "response": "95% CI ≈ [120.85, 126.41].", "meta": {"id": "WUA_STAT_25K_010990", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.11, compute E and Var."}]}, "response": "E[X]=3.85; Var[X]=3.427.", "meta": {"id": "WUA_STAT_25K_010991", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.05, μ=97.15, σ=20.92, compute z."}]}, "response": "z = -2.968.", "meta": {"id": "WUA_STAT_25K_010992", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.14, s=3.53, n=159, compute CI."}]}, "response": "95% CI ≈ [175.59, 176.69].", "meta": {"id": "WUA_STAT_25K_010993", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.54, s=23.86, n=42, compute CI."}]}, "response": "95% CI ≈ [191.32, 205.76].", "meta": {"id": "WUA_STAT_25K_010994", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.27, s=17.7, n=104, compute CI."}]}, "response": "95% CI ≈ [181.87, 188.67].", "meta": {"id": "WUA_STAT_25K_010995", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.7, μ=28.5, σ=7.13, compute z."}]}, "response": "z = -1.935.", "meta": {"id": "WUA_STAT_25K_010996", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.81, μ=67.31, σ=13.82, compute z."}]}, "response": "z = -4.667.", "meta": {"id": "WUA_STAT_25K_010997", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.56, μ=17.51, σ=19.29, compute z."}]}, "response": "z = 3.165.", "meta": {"id": "WUA_STAT_25K_010998", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.9, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.462.", "meta": {"id": "WUA_STAT_25K_010999", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.97, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.879.", "meta": {"id": "WUA_STAT_25K_011000", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.54, compute E and Var."}]}, "response": "E[X]=9.72; Var[X]=4.471.", "meta": {"id": "WUA_STAT_25K_011001", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.055, sens=0.75, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.304.", "meta": {"id": "WUA_STAT_25K_011002", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.81, μ=7.81, σ=13.85, compute z."}]}, "response": "z = 3.682.", "meta": {"id": "WUA_STAT_25K_011003", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.057, sens=0.83, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.501.", "meta": {"id": "WUA_STAT_25K_011004", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.6, s=23.29, n=137, compute CI."}]}, "response": "95% CI ≈ [76.70, 84.50].", "meta": {"id": "WUA_STAT_25K_011005", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.3, μ=89.81, σ=15.8, compute z."}]}, "response": "z = -4.526.", "meta": {"id": "WUA_STAT_25K_011006", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.56, compute E and Var."}]}, "response": "E[X]=8.96; Var[X]=3.942.", "meta": {"id": "WUA_STAT_25K_011007", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.58, compute E and Var."}]}, "response": "E[X]=15.66; Var[X]=6.577.", "meta": {"id": "WUA_STAT_25K_011008", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.49, compute E and Var."}]}, "response": "E[X]=9.80; Var[X]=4.998.", "meta": {"id": "WUA_STAT_25K_011009", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.75, compute E and Var."}]}, "response": "E[X]=17.25; Var[X]=4.312.", "meta": {"id": "WUA_STAT_25K_011010", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.84, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.955.", "meta": {"id": "WUA_STAT_25K_011011", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.233, sens=0.98, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.810.", "meta": {"id": "WUA_STAT_25K_011012", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.39, compute E and Var."}]}, "response": "E[X]=7.80; Var[X]=4.758.", "meta": {"id": "WUA_STAT_25K_011013", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.88, s=7.72, n=84, compute CI."}]}, "response": "95% CI ≈ [66.23, 69.53].", "meta": {"id": "WUA_STAT_25K_011014", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.159, sens=0.75, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.353.", "meta": {"id": "WUA_STAT_25K_011015", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=82.76, s=21.29, n=90, compute CI."}]}, "response": "95% CI ≈ [78.36, 87.16].", "meta": {"id": "WUA_STAT_25K_011016", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.47, compute E and Var."}]}, "response": "E[X]=12.22; Var[X]=6.477.", "meta": {"id": "WUA_STAT_25K_011017", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.72, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.903.", "meta": {"id": "WUA_STAT_25K_011018", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.184, sens=0.73, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.451.", "meta": {"id": "WUA_STAT_25K_011019", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.39, s=2.9, n=166, compute CI."}]}, "response": "95% CI ≈ [13.95, 14.83].", "meta": {"id": "WUA_STAT_25K_011020", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.171, sens=0.92, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.432.", "meta": {"id": "WUA_STAT_25K_011021", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.53, μ=77.63, σ=10.49, compute z."}]}, "response": "z = 1.325.", "meta": {"id": "WUA_STAT_25K_011022", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.232, sens=0.78, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.627.", "meta": {"id": "WUA_STAT_25K_011023", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.51, compute E and Var."}]}, "response": "E[X]=7.65; Var[X]=3.748.", "meta": {"id": "WUA_STAT_25K_011024", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.96, s=25.9, n=153, compute CI."}]}, "response": "95% CI ≈ [93.86, 102.06].", "meta": {"id": "WUA_STAT_25K_011025", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.31, compute E and Var."}]}, "response": "E[X]=2.79; Var[X]=1.925.", "meta": {"id": "WUA_STAT_25K_011026", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.77, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.209.", "meta": {"id": "WUA_STAT_25K_011027", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.187, sens=0.82, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.485.", "meta": {"id": "WUA_STAT_25K_011028", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.61, s=22.29, n=13, compute CI."}]}, "response": "95% CI ≈ [5.49, 29.73].", "meta": {"id": "WUA_STAT_25K_011029", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.94, μ=5.94, σ=5.19, compute z."}]}, "response": "z = 14.451.", "meta": {"id": "WUA_STAT_25K_011030", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.77, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.219.", "meta": {"id": "WUA_STAT_25K_011031", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.73, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.651.", "meta": {"id": "WUA_STAT_25K_011032", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.73, compute E and Var."}]}, "response": "E[X]=16.79; Var[X]=4.533.", "meta": {"id": "WUA_STAT_25K_011033", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.77, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.734.", "meta": {"id": "WUA_STAT_25K_011034", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.9, s=39.46, n=154, compute CI."}]}, "response": "95% CI ≈ [192.67, 205.13].", "meta": {"id": "WUA_STAT_25K_011035", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.09, s=32.45, n=173, compute CI."}]}, "response": "95% CI ≈ [55.25, 64.93].", "meta": {"id": "WUA_STAT_25K_011036", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.046, sens=0.91, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.305.", "meta": {"id": "WUA_STAT_25K_011037", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.89, s=1.12, n=15, compute CI."}]}, "response": "95% CI ≈ [45.32, 46.46].", "meta": {"id": "WUA_STAT_25K_011038", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.42, compute E and Var."}]}, "response": "E[X]=10.92; Var[X]=6.334.", "meta": {"id": "WUA_STAT_25K_011039", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.68, s=38.48, n=49, compute CI."}]}, "response": "95% CI ≈ [43.91, 65.45].", "meta": {"id": "WUA_STAT_25K_011040", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.26, compute E and Var."}]}, "response": "E[X]=9.88; Var[X]=7.311.", "meta": {"id": "WUA_STAT_25K_011041", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.69, compute E and Var."}]}, "response": "E[X]=8.28; Var[X]=2.567.", "meta": {"id": "WUA_STAT_25K_011042", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.228, sens=0.81, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.585.", "meta": {"id": "WUA_STAT_25K_011043", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.87, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.526.", "meta": {"id": "WUA_STAT_25K_011044", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.167, sens=0.76, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.656.", "meta": {"id": "WUA_STAT_25K_011045", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.31, μ=99.92, σ=6.76, compute z."}]}, "response": "z = -2.161.", "meta": {"id": "WUA_STAT_25K_011046", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.32, compute E and Var."}]}, "response": "E[X]=5.44; Var[X]=3.699.", "meta": {"id": "WUA_STAT_25K_011047", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.04, μ=49.03, σ=7.34, compute z."}]}, "response": "z = -3.405.", "meta": {"id": "WUA_STAT_25K_011048", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.14, s=9.58, n=36, compute CI."}]}, "response": "95% CI ≈ [174.01, 180.27].", "meta": {"id": "WUA_STAT_25K_011049", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.205, sens=0.89, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.657.", "meta": {"id": "WUA_STAT_25K_011050", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.6, μ=46.0, σ=21.4, compute z."}]}, "response": "z = 1.243.", "meta": {"id": "WUA_STAT_25K_011051", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.73, μ=76.14, σ=12.44, compute z."}]}, "response": "z = 0.289.", "meta": {"id": "WUA_STAT_25K_011052", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.019, sens=0.74, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.125.", "meta": {"id": "WUA_STAT_25K_011053", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.59, compute E and Var."}]}, "response": "E[X]=17.70; Var[X]=7.257.", "meta": {"id": "WUA_STAT_25K_011054", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.81, μ=90.78, σ=10.69, compute z."}]}, "response": "z = -3.833.", "meta": {"id": "WUA_STAT_25K_011055", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.73, s=28.19, n=116, compute CI."}]}, "response": "95% CI ≈ [173.60, 183.86].", "meta": {"id": "WUA_STAT_25K_011056", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.19, compute E and Var."}]}, "response": "E[X]=4.56; Var[X]=3.694.", "meta": {"id": "WUA_STAT_25K_011057", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.75, compute E and Var."}]}, "response": "E[X]=15.75; Var[X]=3.938.", "meta": {"id": "WUA_STAT_25K_011058", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.53, μ=38.52, σ=14.27, compute z."}]}, "response": "z = 0.491.", "meta": {"id": "WUA_STAT_25K_011059", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.157, sens=0.99, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.586.", "meta": {"id": "WUA_STAT_25K_011060", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.169, sens=0.74, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.537.", "meta": {"id": "WUA_STAT_25K_011061", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=143.66, s=34.8, n=92, compute CI."}]}, "response": "95% CI ≈ [136.55, 150.77].", "meta": {"id": "WUA_STAT_25K_011062", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.1, s=31.5, n=81, compute CI."}]}, "response": "95% CI ≈ [5.24, 18.96].", "meta": {"id": "WUA_STAT_25K_011063", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.35, μ=82.94, σ=6.26, compute z."}]}, "response": "z = -4.088.", "meta": {"id": "WUA_STAT_25K_011064", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.87, μ=75.94, σ=11.86, compute z."}]}, "response": "z = -0.175.", "meta": {"id": "WUA_STAT_25K_011065", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.62, compute E and Var."}]}, "response": "E[X]=21.70; Var[X]=8.246.", "meta": {"id": "WUA_STAT_25K_011066", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.25, μ=3.84, σ=11.6, compute z."}]}, "response": "z = 0.380.", "meta": {"id": "WUA_STAT_25K_011067", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.64, compute E and Var."}]}, "response": "E[X]=8.32; Var[X]=2.995.", "meta": {"id": "WUA_STAT_25K_011068", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.89, μ=67.12, σ=3.35, compute z."}]}, "response": "z = -18.875.", "meta": {"id": "WUA_STAT_25K_011069", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.94, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.369.", "meta": {"id": "WUA_STAT_25K_011070", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.8, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.379.", "meta": {"id": "WUA_STAT_25K_011071", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.11, compute E and Var."}]}, "response": "E[X]=0.99; Var[X]=0.881.", "meta": {"id": "WUA_STAT_25K_011072", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=82.0, s=23.77, n=128, compute CI."}]}, "response": "95% CI ≈ [77.88, 86.12].", "meta": {"id": "WUA_STAT_25K_011073", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.3, compute E and Var."}]}, "response": "E[X]=4.50; Var[X]=3.150.", "meta": {"id": "WUA_STAT_25K_011074", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.3, compute E and Var."}]}, "response": "E[X]=5.40; Var[X]=3.780.", "meta": {"id": "WUA_STAT_25K_011075", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.58, compute E and Var."}]}, "response": "E[X]=2.90; Var[X]=1.218.", "meta": {"id": "WUA_STAT_25K_011076", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.69, s=37.95, n=122, compute CI."}]}, "response": "95% CI ≈ [102.96, 116.42].", "meta": {"id": "WUA_STAT_25K_011077", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.39, s=30.27, n=163, compute CI."}]}, "response": "95% CI ≈ [193.74, 203.04].", "meta": {"id": "WUA_STAT_25K_011078", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.96, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.509.", "meta": {"id": "WUA_STAT_25K_011079", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.71, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.699.", "meta": {"id": "WUA_STAT_25K_011080", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.9, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.789.", "meta": {"id": "WUA_STAT_25K_011081", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.101, sens=0.78, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.276.", "meta": {"id": "WUA_STAT_25K_011082", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.62, compute E and Var."}]}, "response": "E[X]=6.20; Var[X]=2.356.", "meta": {"id": "WUA_STAT_25K_011083", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=128.74, s=31.73, n=31, compute CI."}]}, "response": "95% CI ≈ [117.57, 139.91].", "meta": {"id": "WUA_STAT_25K_011084", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=19.72, s=13.56, n=198, compute CI."}]}, "response": "95% CI ≈ [17.83, 21.61].", "meta": {"id": "WUA_STAT_25K_011085", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.55, compute E and Var."}]}, "response": "E[X]=13.75; Var[X]=6.188.", "meta": {"id": "WUA_STAT_25K_011086", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.88, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.867.", "meta": {"id": "WUA_STAT_25K_011087", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.72, μ=72.94, σ=1.1, compute z."}]}, "response": "z = 5.255.", "meta": {"id": "WUA_STAT_25K_011088", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.1, compute E and Var."}]}, "response": "E[X]=1.10; Var[X]=0.990.", "meta": {"id": "WUA_STAT_25K_011089", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.23, s=27.09, n=169, compute CI."}]}, "response": "95% CI ≈ [146.15, 154.31].", "meta": {"id": "WUA_STAT_25K_011090", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.17, compute E and Var."}]}, "response": "E[X]=1.19; Var[X]=0.988.", "meta": {"id": "WUA_STAT_25K_011091", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.73, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.045.", "meta": {"id": "WUA_STAT_25K_011092", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.31, compute E and Var."}]}, "response": "E[X]=8.06; Var[X]=5.561.", "meta": {"id": "WUA_STAT_25K_011093", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.027, sens=0.95, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.169.", "meta": {"id": "WUA_STAT_25K_011094", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.24, s=18.14, n=58, compute CI."}]}, "response": "95% CI ≈ [141.57, 150.91].", "meta": {"id": "WUA_STAT_25K_011095", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.23, s=22.98, n=131, compute CI."}]}, "response": "95% CI ≈ [103.29, 111.17].", "meta": {"id": "WUA_STAT_25K_011096", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=10.42, s=19.97, n=26, compute CI."}]}, "response": "95% CI ≈ [2.74, 18.10].", "meta": {"id": "WUA_STAT_25K_011097", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.76, μ=24.36, σ=10.81, compute z."}]}, "response": "z = -0.518.", "meta": {"id": "WUA_STAT_25K_011098", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=32.81, s=6.46, n=91, compute CI."}]}, "response": "95% CI ≈ [31.48, 34.14].", "meta": {"id": "WUA_STAT_25K_011099", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.25, sens=0.93, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.525.", "meta": {"id": "WUA_STAT_25K_011100", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.75, μ=6.88, σ=15.05, compute z."}]}, "response": "z = 1.520.", "meta": {"id": "WUA_STAT_25K_011101", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.21, sens=0.87, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.623.", "meta": {"id": "WUA_STAT_25K_011102", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.077, sens=0.96, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.297.", "meta": {"id": "WUA_STAT_25K_011103", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.44, compute E and Var."}]}, "response": "E[X]=16.28; Var[X]=9.117.", "meta": {"id": "WUA_STAT_25K_011104", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.89, μ=59.78, σ=16.97, compute z."}]}, "response": "z = -0.406.", "meta": {"id": "WUA_STAT_25K_011105", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.39, μ=51.1, σ=17.12, compute z."}]}, "response": "z = 0.659.", "meta": {"id": "WUA_STAT_25K_011106", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=10.4, s=4.19, n=26, compute CI."}]}, "response": "95% CI ≈ [8.79, 12.01].", "meta": {"id": "WUA_STAT_25K_011107", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.197, sens=0.9, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.580.", "meta": {"id": "WUA_STAT_25K_011108", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=108.89, s=39.68, n=24, compute CI."}]}, "response": "95% CI ≈ [93.01, 124.77].", "meta": {"id": "WUA_STAT_25K_011109", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.78, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.382.", "meta": {"id": "WUA_STAT_25K_011110", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.42, s=24.01, n=54, compute CI."}]}, "response": "95% CI ≈ [146.02, 158.82].", "meta": {"id": "WUA_STAT_25K_011111", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.5, s=26.45, n=66, compute CI."}]}, "response": "95% CI ≈ [29.12, 41.88].", "meta": {"id": "WUA_STAT_25K_011112", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.29, compute E and Var."}]}, "response": "E[X]=6.38; Var[X]=4.530.", "meta": {"id": "WUA_STAT_25K_011113", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.019, sens=0.9, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.199.", "meta": {"id": "WUA_STAT_25K_011114", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.184, sens=0.84, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.863.", "meta": {"id": "WUA_STAT_25K_011115", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.53, μ=60.87, σ=23.75, compute z."}]}, "response": "z = 1.417.", "meta": {"id": "WUA_STAT_25K_011116", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.062, sens=0.93, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.266.", "meta": {"id": "WUA_STAT_25K_011117", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.37, compute E and Var."}]}, "response": "E[X]=9.62; Var[X]=6.061.", "meta": {"id": "WUA_STAT_25K_011118", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.4, μ=64.14, σ=5.5, compute z."}]}, "response": "z = -9.771.", "meta": {"id": "WUA_STAT_25K_011119", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.27, μ=32.18, σ=5.91, compute z."}]}, "response": "z = 5.937.", "meta": {"id": "WUA_STAT_25K_011120", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.18, s=17.18, n=54, compute CI."}]}, "response": "95% CI ≈ [161.60, 170.76].", "meta": {"id": "WUA_STAT_25K_011121", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.72, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.592.", "meta": {"id": "WUA_STAT_25K_011122", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.75, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.662.", "meta": {"id": "WUA_STAT_25K_011123", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=20.54, s=28.67, n=54, compute CI."}]}, "response": "95% CI ≈ [12.89, 28.19].", "meta": {"id": "WUA_STAT_25K_011124", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.7, s=4.65, n=155, compute CI."}]}, "response": "95% CI ≈ [128.97, 130.43].", "meta": {"id": "WUA_STAT_25K_011125", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.71, μ=59.08, σ=13.75, compute z."}]}, "response": "z = -4.027.", "meta": {"id": "WUA_STAT_25K_011126", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.42, μ=39.43, σ=14.38, compute z."}]}, "response": "z = 0.069.", "meta": {"id": "WUA_STAT_25K_011127", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.14, compute E and Var."}]}, "response": "E[X]=4.90; Var[X]=4.214.", "meta": {"id": "WUA_STAT_25K_011128", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.19, s=14.61, n=183, compute CI."}]}, "response": "95% CI ≈ [133.07, 137.31].", "meta": {"id": "WUA_STAT_25K_011129", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.97, s=23.86, n=21, compute CI."}]}, "response": "95% CI ≈ [80.76, 101.18].", "meta": {"id": "WUA_STAT_25K_011130", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.98, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.478.", "meta": {"id": "WUA_STAT_25K_011131", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.82, s=28.66, n=145, compute CI."}]}, "response": "95% CI ≈ [27.16, 36.48].", "meta": {"id": "WUA_STAT_25K_011132", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.45, μ=35.4, σ=22.64, compute z."}]}, "response": "z = -0.219.", "meta": {"id": "WUA_STAT_25K_011133", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.82, μ=24.4, σ=5.16, compute z."}]}, "response": "z = 4.733.", "meta": {"id": "WUA_STAT_25K_011134", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.81, μ=51.22, σ=12.08, compute z."}]}, "response": "z = -1.938.", "meta": {"id": "WUA_STAT_25K_011135", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.73, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.460.", "meta": {"id": "WUA_STAT_25K_011136", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=104.81, s=11.21, n=160, compute CI."}]}, "response": "95% CI ≈ [103.07, 106.55].", "meta": {"id": "WUA_STAT_25K_011137", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.14, compute E and Var."}]}, "response": "E[X]=4.34; Var[X]=3.732.", "meta": {"id": "WUA_STAT_25K_011138", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.6, compute E and Var."}]}, "response": "E[X]=20.40; Var[X]=8.160.", "meta": {"id": "WUA_STAT_25K_011139", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.68, s=26.17, n=80, compute CI."}]}, "response": "95% CI ≈ [43.95, 55.41].", "meta": {"id": "WUA_STAT_25K_011140", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.12, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=2.218.", "meta": {"id": "WUA_STAT_25K_011141", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.96, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.241.", "meta": {"id": "WUA_STAT_25K_011142", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.41, μ=25.28, σ=1.37, compute z."}]}, "response": "z = 52.650.", "meta": {"id": "WUA_STAT_25K_011143", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.71, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.260.", "meta": {"id": "WUA_STAT_25K_011144", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.82, μ=60.41, σ=4.21, compute z."}]}, "response": "z = -13.204.", "meta": {"id": "WUA_STAT_25K_011145", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.79, compute E and Var."}]}, "response": "E[X]=31.60; Var[X]=6.636.", "meta": {"id": "WUA_STAT_25K_011146", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=164.38, s=21.93, n=88, compute CI."}]}, "response": "95% CI ≈ [159.80, 168.96].", "meta": {"id": "WUA_STAT_25K_011147", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.66, s=29.8, n=85, compute CI."}]}, "response": "95% CI ≈ [15.32, 28.00].", "meta": {"id": "WUA_STAT_25K_011148", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.42, s=39.01, n=123, compute CI."}]}, "response": "95% CI ≈ [115.53, 129.31].", "meta": {"id": "WUA_STAT_25K_011149", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.26, μ=90.12, σ=16.12, compute z."}]}, "response": "z = -4.520.", "meta": {"id": "WUA_STAT_25K_011150", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.83, μ=20.82, σ=5.73, compute z."}]}, "response": "z = -3.489.", "meta": {"id": "WUA_STAT_25K_011151", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.65, μ=12.46, σ=19.37, compute z."}]}, "response": "z = 0.733.", "meta": {"id": "WUA_STAT_25K_011152", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=91.74, s=1.86, n=121, compute CI."}]}, "response": "95% CI ≈ [91.41, 92.07].", "meta": {"id": "WUA_STAT_25K_011153", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.212, sens=0.82, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.612.", "meta": {"id": "WUA_STAT_25K_011154", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.52, compute E and Var."}]}, "response": "E[X]=3.64; Var[X]=1.747.", "meta": {"id": "WUA_STAT_25K_011155", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.75, compute E and Var."}]}, "response": "E[X]=5.25; Var[X]=1.312.", "meta": {"id": "WUA_STAT_25K_011156", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.07, μ=1.85, σ=3.46, compute z."}]}, "response": "z = 14.803.", "meta": {"id": "WUA_STAT_25K_011157", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.29, μ=74.86, σ=21.21, compute z."}]}, "response": "z = -0.923.", "meta": {"id": "WUA_STAT_25K_011158", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.5, compute E and Var."}]}, "response": "E[X]=18.50; Var[X]=9.250.", "meta": {"id": "WUA_STAT_25K_011159", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.86, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.380.", "meta": {"id": "WUA_STAT_25K_011160", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.024, sens=0.98, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.156.", "meta": {"id": "WUA_STAT_25K_011161", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.91, μ=92.56, σ=5.9, compute z."}]}, "response": "z = -6.890.", "meta": {"id": "WUA_STAT_25K_011162", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.027, sens=0.89, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.331.", "meta": {"id": "WUA_STAT_25K_011163", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.53, compute E and Var."}]}, "response": "E[X]=15.37; Var[X]=7.224.", "meta": {"id": "WUA_STAT_25K_011164", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.73, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.658.", "meta": {"id": "WUA_STAT_25K_011165", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.35, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.365.", "meta": {"id": "WUA_STAT_25K_011166", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.97, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.485.", "meta": {"id": "WUA_STAT_25K_011167", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.92, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.498.", "meta": {"id": "WUA_STAT_25K_011168", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.17, compute E and Var."}]}, "response": "E[X]=4.08; Var[X]=3.386.", "meta": {"id": "WUA_STAT_25K_011169", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.73, compute E and Var."}]}, "response": "E[X]=10.22; Var[X]=2.759.", "meta": {"id": "WUA_STAT_25K_011170", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.28, μ=72.42, σ=19.27, compute z."}]}, "response": "z = -1.875.", "meta": {"id": "WUA_STAT_25K_011171", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.88, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.117.", "meta": {"id": "WUA_STAT_25K_011172", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.76, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.697.", "meta": {"id": "WUA_STAT_25K_011173", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=126.41, s=16.7, n=51, compute CI."}]}, "response": "95% CI ≈ [121.83, 130.99].", "meta": {"id": "WUA_STAT_25K_011174", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.063, sens=0.72, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.162.", "meta": {"id": "WUA_STAT_25K_011175", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.67, μ=76.69, σ=21.1, compute z."}]}, "response": "z = -1.755.", "meta": {"id": "WUA_STAT_25K_011176", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.49, compute E and Var."}]}, "response": "E[X]=10.78; Var[X]=5.498.", "meta": {"id": "WUA_STAT_25K_011177", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.77, μ=30.87, σ=13.87, compute z."}]}, "response": "z = 2.444.", "meta": {"id": "WUA_STAT_25K_011178", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.71, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.280.", "meta": {"id": "WUA_STAT_25K_011179", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.13, μ=4.32, σ=15.37, compute z."}]}, "response": "z = 3.241.", "meta": {"id": "WUA_STAT_25K_011180", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.88, μ=77.93, σ=7.8, compute z."}]}, "response": "z = -3.083.", "meta": {"id": "WUA_STAT_25K_011181", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.213, sens=0.84, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.448.", "meta": {"id": "WUA_STAT_25K_011182", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.48, compute E and Var."}]}, "response": "E[X]=5.76; Var[X]=2.995.", "meta": {"id": "WUA_STAT_25K_011183", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.52, compute E and Var."}]}, "response": "E[X]=4.68; Var[X]=2.246.", "meta": {"id": "WUA_STAT_25K_011184", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.22, s=11.08, n=51, compute CI."}]}, "response": "95% CI ≈ [160.18, 166.26].", "meta": {"id": "WUA_STAT_25K_011185", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.53, s=30.0, n=131, compute CI."}]}, "response": "95% CI ≈ [177.39, 187.67].", "meta": {"id": "WUA_STAT_25K_011186", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.89, μ=77.68, σ=8.06, compute z."}]}, "response": "z = -4.192.", "meta": {"id": "WUA_STAT_25K_011187", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=95.47, s=18.29, n=81, compute CI."}]}, "response": "95% CI ≈ [91.49, 99.45].", "meta": {"id": "WUA_STAT_25K_011188", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=28.61, s=4.23, n=75, compute CI."}]}, "response": "95% CI ≈ [27.65, 29.57].", "meta": {"id": "WUA_STAT_25K_011189", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.213, sens=0.94, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.525.", "meta": {"id": "WUA_STAT_25K_011190", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.185, sens=0.84, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.704.", "meta": {"id": "WUA_STAT_25K_011191", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.6, compute E and Var."}]}, "response": "E[X]=18.00; Var[X]=7.200.", "meta": {"id": "WUA_STAT_25K_011192", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.46, compute E and Var."}]}, "response": "E[X]=12.42; Var[X]=6.707.", "meta": {"id": "WUA_STAT_25K_011193", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.68, μ=97.7, σ=9.35, compute z."}]}, "response": "z = -5.778.", "meta": {"id": "WUA_STAT_25K_011194", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.44, compute E and Var."}]}, "response": "E[X]=13.64; Var[X]=7.638.", "meta": {"id": "WUA_STAT_25K_011195", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.35, compute E and Var."}]}, "response": "E[X]=12.95; Var[X]=8.418.", "meta": {"id": "WUA_STAT_25K_011196", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.48, s=37.41, n=120, compute CI."}]}, "response": "95% CI ≈ [170.79, 184.17].", "meta": {"id": "WUA_STAT_25K_011197", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.35, s=11.2, n=27, compute CI."}]}, "response": "95% CI ≈ [109.13, 117.57].", "meta": {"id": "WUA_STAT_25K_011198", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.81, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.220.", "meta": {"id": "WUA_STAT_25K_011199", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.55, μ=3.52, σ=1.47, compute z."}]}, "response": "z = 31.313.", "meta": {"id": "WUA_STAT_25K_011200", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.01, s=29.99, n=38, compute CI."}]}, "response": "95% CI ≈ [169.47, 188.55].", "meta": {"id": "WUA_STAT_25K_011201", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.58, μ=32.52, σ=17.83, compute z."}]}, "response": "z = 1.742.", "meta": {"id": "WUA_STAT_25K_011202", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.211, sens=0.87, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.769.", "meta": {"id": "WUA_STAT_25K_011203", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.153, sens=0.74, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.378.", "meta": {"id": "WUA_STAT_25K_011204", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.36, s=12.5, n=110, compute CI."}]}, "response": "95% CI ≈ [86.02, 90.70].", "meta": {"id": "WUA_STAT_25K_011205", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.6, s=1.86, n=49, compute CI."}]}, "response": "95% CI ≈ [188.08, 189.12].", "meta": {"id": "WUA_STAT_25K_011206", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.08, compute E and Var."}]}, "response": "E[X]=2.24; Var[X]=2.061.", "meta": {"id": "WUA_STAT_25K_011207", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.75, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.279.", "meta": {"id": "WUA_STAT_25K_011208", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.42, s=9.45, n=125, compute CI."}]}, "response": "95% CI ≈ [13.76, 17.08].", "meta": {"id": "WUA_STAT_25K_011209", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.63, compute E and Var."}]}, "response": "E[X]=20.16; Var[X]=7.459.", "meta": {"id": "WUA_STAT_25K_011210", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.53, μ=74.4, σ=2.64, compute z."}]}, "response": "z = -10.557.", "meta": {"id": "WUA_STAT_25K_011211", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.51, s=2.25, n=186, compute CI."}]}, "response": "95% CI ≈ [187.19, 187.83].", "meta": {"id": "WUA_STAT_25K_011212", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.44, s=10.01, n=125, compute CI."}]}, "response": "95% CI ≈ [115.69, 119.19].", "meta": {"id": "WUA_STAT_25K_011213", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.92, s=14.98, n=18, compute CI."}]}, "response": "95% CI ≈ [9.00, 22.84].", "meta": {"id": "WUA_STAT_25K_011214", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.4, s=10.84, n=106, compute CI."}]}, "response": "95% CI ≈ [78.34, 82.46].", "meta": {"id": "WUA_STAT_25K_011215", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.23, s=9.46, n=118, compute CI."}]}, "response": "95% CI ≈ [177.52, 180.94].", "meta": {"id": "WUA_STAT_25K_011216", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.2, sens=0.82, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.471.", "meta": {"id": "WUA_STAT_25K_011217", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.89, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.967.", "meta": {"id": "WUA_STAT_25K_011218", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.25, compute E and Var."}]}, "response": "E[X]=6.25; Var[X]=4.688.", "meta": {"id": "WUA_STAT_25K_011219", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.41, s=38.25, n=30, compute CI."}]}, "response": "95% CI ≈ [147.72, 175.10].", "meta": {"id": "WUA_STAT_25K_011220", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.22, compute E and Var."}]}, "response": "E[X]=3.30; Var[X]=2.574.", "meta": {"id": "WUA_STAT_25K_011221", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.117, sens=0.9, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.412.", "meta": {"id": "WUA_STAT_25K_011222", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.61, μ=26.77, σ=18.11, compute z."}]}, "response": "z = 2.918.", "meta": {"id": "WUA_STAT_25K_011223", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=168.43, s=29.63, n=168, compute CI."}]}, "response": "95% CI ≈ [163.95, 172.91].", "meta": {"id": "WUA_STAT_25K_011224", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.59, μ=39.89, σ=17.47, compute z."}]}, "response": "z = 2.387.", "meta": {"id": "WUA_STAT_25K_011225", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.65, compute E and Var."}]}, "response": "E[X]=21.45; Var[X]=7.507.", "meta": {"id": "WUA_STAT_25K_011226", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.72, μ=31.03, σ=7.08, compute z."}]}, "response": "z = 8.007.", "meta": {"id": "WUA_STAT_25K_011227", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.37, μ=63.25, σ=13.08, compute z."}]}, "response": "z = 1.462.", "meta": {"id": "WUA_STAT_25K_011228", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=102.28, s=9.4, n=134, compute CI."}]}, "response": "95% CI ≈ [100.69, 103.87].", "meta": {"id": "WUA_STAT_25K_011229", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.8, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.525.", "meta": {"id": "WUA_STAT_25K_011230", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.26, compute E and Var."}]}, "response": "E[X]=4.68; Var[X]=3.463.", "meta": {"id": "WUA_STAT_25K_011231", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.1, μ=71.94, σ=12.32, compute z."}]}, "response": "z = -4.045.", "meta": {"id": "WUA_STAT_25K_011232", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.31, s=7.04, n=95, compute CI."}]}, "response": "95% CI ≈ [48.89, 51.73].", "meta": {"id": "WUA_STAT_25K_011233", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.04, μ=97.57, σ=11.4, compute z."}]}, "response": "z = -0.924.", "meta": {"id": "WUA_STAT_25K_011234", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.04, s=34.13, n=103, compute CI."}]}, "response": "95% CI ≈ [90.45, 103.63].", "meta": {"id": "WUA_STAT_25K_011235", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.41, μ=60.72, σ=11.45, compute z."}]}, "response": "z = -2.822.", "meta": {"id": "WUA_STAT_25K_011236", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.45, s=12.98, n=38, compute CI."}]}, "response": "95% CI ≈ [55.32, 63.58].", "meta": {"id": "WUA_STAT_25K_011237", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.8, s=34.83, n=200, compute CI."}]}, "response": "95% CI ≈ [165.97, 175.63].", "meta": {"id": "WUA_STAT_25K_011238", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.24, compute E and Var."}]}, "response": "E[X]=6.48; Var[X]=4.925.", "meta": {"id": "WUA_STAT_25K_011239", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.15, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.785.", "meta": {"id": "WUA_STAT_25K_011240", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.217, sens=0.77, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.461.", "meta": {"id": "WUA_STAT_25K_011241", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.91, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.623.", "meta": {"id": "WUA_STAT_25K_011242", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.55, compute E and Var."}]}, "response": "E[X]=4.95; Var[X]=2.228.", "meta": {"id": "WUA_STAT_25K_011243", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.28, μ=52.03, σ=7.95, compute z."}]}, "response": "z = 0.535.", "meta": {"id": "WUA_STAT_25K_011244", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.91, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.382.", "meta": {"id": "WUA_STAT_25K_011245", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.37, s=37.84, n=93, compute CI."}]}, "response": "95% CI ≈ [103.68, 119.06].", "meta": {"id": "WUA_STAT_25K_011246", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.67, compute E and Var."}]}, "response": "E[X]=21.44; Var[X]=7.075.", "meta": {"id": "WUA_STAT_25K_011247", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.1, μ=95.96, σ=5.36, compute z."}]}, "response": "z = -16.019.", "meta": {"id": "WUA_STAT_25K_011248", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.87, μ=46.72, σ=13.93, compute z."}]}, "response": "z = 3.313.", "meta": {"id": "WUA_STAT_25K_011249", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.78, compute E and Var."}]}, "response": "E[X]=16.38; Var[X]=3.604.", "meta": {"id": "WUA_STAT_25K_011250", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.55, s=15.59, n=117, compute CI."}]}, "response": "95% CI ≈ [142.73, 148.37].", "meta": {"id": "WUA_STAT_25K_011251", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.56, μ=44.79, σ=24.4, compute z."}]}, "response": "z = -1.321.", "meta": {"id": "WUA_STAT_25K_011252", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.8, μ=87.06, σ=2.87, compute z."}]}, "response": "z = -23.436.", "meta": {"id": "WUA_STAT_25K_011253", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.11, μ=85.67, σ=9.1, compute z."}]}, "response": "z = -8.853.", "meta": {"id": "WUA_STAT_25K_011254", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.0, μ=92.93, σ=20.86, compute z."}]}, "response": "z = -0.045.", "meta": {"id": "WUA_STAT_25K_011255", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.26, s=30.63, n=13, compute CI."}]}, "response": "95% CI ≈ [8.61, 41.91].", "meta": {"id": "WUA_STAT_25K_011256", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.83, μ=56.43, σ=23.78, compute z."}]}, "response": "z = -1.413.", "meta": {"id": "WUA_STAT_25K_011257", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.8, μ=1.51, σ=23.37, compute z."}]}, "response": "z = 2.922.", "meta": {"id": "WUA_STAT_25K_011258", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.79, s=31.48, n=117, compute CI."}]}, "response": "95% CI ≈ [137.09, 148.49].", "meta": {"id": "WUA_STAT_25K_011259", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.71, compute E and Var."}]}, "response": "E[X]=17.75; Var[X]=5.148.", "meta": {"id": "WUA_STAT_25K_011260", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.09, s=33.18, n=70, compute CI."}]}, "response": "95% CI ≈ [72.32, 87.86].", "meta": {"id": "WUA_STAT_25K_011261", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.95, s=36.34, n=146, compute CI."}]}, "response": "95% CI ≈ [105.06, 116.84].", "meta": {"id": "WUA_STAT_25K_011262", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.1, μ=49.61, σ=2.99, compute z."}]}, "response": "z = -14.886.", "meta": {"id": "WUA_STAT_25K_011263", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.29, s=6.58, n=127, compute CI."}]}, "response": "95% CI ≈ [118.15, 120.43].", "meta": {"id": "WUA_STAT_25K_011264", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.49, s=12.21, n=16, compute CI."}]}, "response": "95% CI ≈ [31.51, 43.47].", "meta": {"id": "WUA_STAT_25K_011265", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.89, μ=41.25, σ=22.77, compute z."}]}, "response": "z = -0.587.", "meta": {"id": "WUA_STAT_25K_011266", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.06, compute E and Var."}]}, "response": "E[X]=2.34; Var[X]=2.200.", "meta": {"id": "WUA_STAT_25K_011267", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.91, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.658.", "meta": {"id": "WUA_STAT_25K_011268", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.58, compute E and Var."}]}, "response": "E[X]=12.76; Var[X]=5.359.", "meta": {"id": "WUA_STAT_25K_011269", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.35, s=18.61, n=20, compute CI."}]}, "response": "95% CI ≈ [81.19, 97.51].", "meta": {"id": "WUA_STAT_25K_011270", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.74, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.819.", "meta": {"id": "WUA_STAT_25K_011271", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.88, s=6.74, n=197, compute CI."}]}, "response": "95% CI ≈ [10.94, 12.82].", "meta": {"id": "WUA_STAT_25K_011272", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.23, s=5.88, n=71, compute CI."}]}, "response": "95% CI ≈ [59.86, 62.60].", "meta": {"id": "WUA_STAT_25K_011273", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.2, compute E and Var."}]}, "response": "E[X]=1.60; Var[X]=1.280.", "meta": {"id": "WUA_STAT_25K_011274", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.01, s=20.06, n=16, compute CI."}]}, "response": "95% CI ≈ [8.18, 27.84].", "meta": {"id": "WUA_STAT_25K_011275", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.78, μ=37.52, σ=20.37, compute z."}]}, "response": "z = 2.664.", "meta": {"id": "WUA_STAT_25K_011276", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.05, μ=92.56, σ=21.22, compute z."}]}, "response": "z = -1.438.", "meta": {"id": "WUA_STAT_25K_011277", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.83, μ=68.03, σ=24.34, compute z."}]}, "response": "z = -0.953.", "meta": {"id": "WUA_STAT_25K_011278", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.097, sens=0.85, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.313.", "meta": {"id": "WUA_STAT_25K_011279", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.121, sens=0.75, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.378.", "meta": {"id": "WUA_STAT_25K_011280", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.8, μ=40.0, σ=18.92, compute z."}]}, "response": "z = 2.104.", "meta": {"id": "WUA_STAT_25K_011281", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=147.28, s=10.14, n=168, compute CI."}]}, "response": "95% CI ≈ [145.75, 148.81].", "meta": {"id": "WUA_STAT_25K_011282", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.187, sens=0.98, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.464.", "meta": {"id": "WUA_STAT_25K_011283", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.58, s=29.5, n=34, compute CI."}]}, "response": "95% CI ≈ [169.66, 189.50].", "meta": {"id": "WUA_STAT_25K_011284", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.18, s=9.76, n=94, compute CI."}]}, "response": "95% CI ≈ [132.21, 136.15].", "meta": {"id": "WUA_STAT_25K_011285", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.4, μ=54.19, σ=10.39, compute z."}]}, "response": "z = 4.351.", "meta": {"id": "WUA_STAT_25K_011286", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.02, μ=92.51, σ=5.82, compute z."}]}, "response": "z = -13.314.", "meta": {"id": "WUA_STAT_25K_011287", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.38, μ=38.75, σ=13.26, compute z."}]}, "response": "z = 2.762.", "meta": {"id": "WUA_STAT_25K_011288", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.47, μ=20.01, σ=2.13, compute z."}]}, "response": "z = -5.418.", "meta": {"id": "WUA_STAT_25K_011289", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.88, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.388.", "meta": {"id": "WUA_STAT_25K_011290", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.64, s=13.07, n=189, compute CI."}]}, "response": "95% CI ≈ [82.78, 86.50].", "meta": {"id": "WUA_STAT_25K_011291", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.43, compute E and Var."}]}, "response": "E[X]=11.61; Var[X]=6.618.", "meta": {"id": "WUA_STAT_25K_011292", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.58, compute E and Var."}]}, "response": "E[X]=5.22; Var[X]=2.192.", "meta": {"id": "WUA_STAT_25K_011293", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=71.42, s=6.8, n=36, compute CI."}]}, "response": "95% CI ≈ [69.20, 73.64].", "meta": {"id": "WUA_STAT_25K_011294", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.25, s=25.81, n=103, compute CI."}]}, "response": "95% CI ≈ [38.27, 48.23].", "meta": {"id": "WUA_STAT_25K_011295", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.56, compute E and Var."}]}, "response": "E[X]=15.68; Var[X]=6.899.", "meta": {"id": "WUA_STAT_25K_011296", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.36, compute E and Var."}]}, "response": "E[X]=11.52; Var[X]=7.373.", "meta": {"id": "WUA_STAT_25K_011297", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.97, s=37.42, n=121, compute CI."}]}, "response": "95% CI ≈ [151.30, 164.64].", "meta": {"id": "WUA_STAT_25K_011298", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.17, μ=29.09, σ=5.13, compute z."}]}, "response": "z = 0.016.", "meta": {"id": "WUA_STAT_25K_011299", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.12, compute E and Var."}]}, "response": "E[X]=2.76; Var[X]=2.429.", "meta": {"id": "WUA_STAT_25K_011300", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.31, μ=76.59, σ=16.76, compute z."}]}, "response": "z = -3.656.", "meta": {"id": "WUA_STAT_25K_011301", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.067, sens=0.83, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.749.", "meta": {"id": "WUA_STAT_25K_011302", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.32, μ=16.43, σ=13.28, compute z."}]}, "response": "z = -0.084.", "meta": {"id": "WUA_STAT_25K_011303", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.46, μ=98.98, σ=5.26, compute z."}]}, "response": "z = -3.901.", "meta": {"id": "WUA_STAT_25K_011304", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=81.81, s=33.03, n=65, compute CI."}]}, "response": "95% CI ≈ [73.78, 89.84].", "meta": {"id": "WUA_STAT_25K_011305", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.23, s=4.68, n=111, compute CI."}]}, "response": "95% CI ≈ [60.36, 62.10].", "meta": {"id": "WUA_STAT_25K_011306", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.5, compute E and Var."}]}, "response": "E[X]=18.50; Var[X]=9.250.", "meta": {"id": "WUA_STAT_25K_011307", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.31, compute E and Var."}]}, "response": "E[X]=9.61; Var[X]=6.631.", "meta": {"id": "WUA_STAT_25K_011308", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.107, sens=0.93, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.396.", "meta": {"id": "WUA_STAT_25K_011309", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.16, compute E and Var."}]}, "response": "E[X]=4.32; Var[X]=3.629.", "meta": {"id": "WUA_STAT_25K_011310", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.47, compute E and Var."}]}, "response": "E[X]=2.82; Var[X]=1.495.", "meta": {"id": "WUA_STAT_25K_011311", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.21, sens=0.78, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.474.", "meta": {"id": "WUA_STAT_25K_011312", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.246, sens=0.97, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.779.", "meta": {"id": "WUA_STAT_25K_011313", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.2, μ=45.23, σ=21.59, compute z."}]}, "response": "z = -1.345.", "meta": {"id": "WUA_STAT_25K_011314", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.5, s=4.75, n=117, compute CI."}]}, "response": "95% CI ≈ [88.64, 90.36].", "meta": {"id": "WUA_STAT_25K_011315", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.31, compute E and Var."}]}, "response": "E[X]=3.72; Var[X]=2.567.", "meta": {"id": "WUA_STAT_25K_011316", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=168.38, s=35.45, n=137, compute CI."}]}, "response": "95% CI ≈ [162.44, 174.32].", "meta": {"id": "WUA_STAT_25K_011317", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.016, sens=0.74, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.091.", "meta": {"id": "WUA_STAT_25K_011318", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.04, μ=19.57, σ=1.3, compute z."}]}, "response": "z = 53.438.", "meta": {"id": "WUA_STAT_25K_011319", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.91, μ=15.69, σ=3.57, compute z."}]}, "response": "z = 1.742.", "meta": {"id": "WUA_STAT_25K_011320", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.96, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.433.", "meta": {"id": "WUA_STAT_25K_011321", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.12, compute E and Var."}]}, "response": "E[X]=1.68; Var[X]=1.478.", "meta": {"id": "WUA_STAT_25K_011322", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.71, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.345.", "meta": {"id": "WUA_STAT_25K_011323", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.76, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.653.", "meta": {"id": "WUA_STAT_25K_011324", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.77, compute E and Var."}]}, "response": "E[X]=16.94; Var[X]=3.896.", "meta": {"id": "WUA_STAT_25K_011325", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.73, compute E and Var."}]}, "response": "E[X]=26.28; Var[X]=7.096.", "meta": {"id": "WUA_STAT_25K_011326", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.74, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.699.", "meta": {"id": "WUA_STAT_25K_011327", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.14, compute E and Var."}]}, "response": "E[X]=2.94; Var[X]=2.528.", "meta": {"id": "WUA_STAT_25K_011328", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.06, s=30.3, n=184, compute CI."}]}, "response": "95% CI ≈ [151.68, 160.44].", "meta": {"id": "WUA_STAT_25K_011329", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.12, compute E and Var."}]}, "response": "E[X]=3.72; Var[X]=3.274.", "meta": {"id": "WUA_STAT_25K_011330", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.76, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.392.", "meta": {"id": "WUA_STAT_25K_011331", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.23, μ=50.96, σ=13.9, compute z."}]}, "response": "z = 2.609.", "meta": {"id": "WUA_STAT_25K_011332", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.06, s=16.74, n=27, compute CI."}]}, "response": "95% CI ≈ [33.75, 46.37].", "meta": {"id": "WUA_STAT_25K_011333", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.058, sens=0.9, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.357.", "meta": {"id": "WUA_STAT_25K_011334", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.044, sens=0.75, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.136.", "meta": {"id": "WUA_STAT_25K_011335", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.85, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.398.", "meta": {"id": "WUA_STAT_25K_011336", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.37, compute E and Var."}]}, "response": "E[X]=7.03; Var[X]=4.429.", "meta": {"id": "WUA_STAT_25K_011337", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.54, s=27.29, n=97, compute CI."}]}, "response": "95% CI ≈ [194.11, 204.97].", "meta": {"id": "WUA_STAT_25K_011338", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.93, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.498.", "meta": {"id": "WUA_STAT_25K_011339", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.73, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.849.", "meta": {"id": "WUA_STAT_25K_011340", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.21, μ=27.15, σ=4.5, compute z."}]}, "response": "z = 8.458.", "meta": {"id": "WUA_STAT_25K_011341", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.166, sens=0.84, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.432.", "meta": {"id": "WUA_STAT_25K_011342", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.89, s=19.85, n=19, compute CI."}]}, "response": "95% CI ≈ [92.96, 110.82].", "meta": {"id": "WUA_STAT_25K_011343", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.81, s=10.77, n=114, compute CI."}]}, "response": "95% CI ≈ [121.83, 125.79].", "meta": {"id": "WUA_STAT_25K_011344", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.96, μ=22.15, σ=14.58, compute z."}]}, "response": "z = 3.553.", "meta": {"id": "WUA_STAT_25K_011345", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.209, sens=0.95, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.464.", "meta": {"id": "WUA_STAT_25K_011346", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.06, μ=69.8, σ=21.39, compute z."}]}, "response": "z = 0.293.", "meta": {"id": "WUA_STAT_25K_011347", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.21, compute E and Var."}]}, "response": "E[X]=6.51; Var[X]=5.143.", "meta": {"id": "WUA_STAT_25K_011348", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.99, μ=83.45, σ=1.89, compute z."}]}, "response": "z = -2.360.", "meta": {"id": "WUA_STAT_25K_011349", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=169.89, s=15.13, n=102, compute CI."}]}, "response": "95% CI ≈ [166.95, 172.83].", "meta": {"id": "WUA_STAT_25K_011350", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.195, sens=0.93, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.446.", "meta": {"id": "WUA_STAT_25K_011351", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.37, μ=85.9, σ=22.32, compute z."}]}, "response": "z = -2.622.", "meta": {"id": "WUA_STAT_25K_011352", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.133, sens=0.87, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.527.", "meta": {"id": "WUA_STAT_25K_011353", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.46, compute E and Var."}]}, "response": "E[X]=15.18; Var[X]=8.197.", "meta": {"id": "WUA_STAT_25K_011354", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=115.71, s=30.7, n=98, compute CI."}]}, "response": "95% CI ≈ [109.63, 121.79].", "meta": {"id": "WUA_STAT_25K_011355", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.93, μ=47.95, σ=5.27, compute z."}]}, "response": "z = -7.214.", "meta": {"id": "WUA_STAT_25K_011356", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.66, μ=39.71, σ=8.06, compute z."}]}, "response": "z = -4.721.", "meta": {"id": "WUA_STAT_25K_011357", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.9, μ=49.42, σ=23.64, compute z."}]}, "response": "z = 1.078.", "meta": {"id": "WUA_STAT_25K_011358", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.39, s=3.19, n=50, compute CI."}]}, "response": "95% CI ≈ [113.51, 115.27].", "meta": {"id": "WUA_STAT_25K_011359", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.6, s=38.12, n=189, compute CI."}]}, "response": "95% CI ≈ [22.17, 33.03].", "meta": {"id": "WUA_STAT_25K_011360", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.05, sens=0.76, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.286.", "meta": {"id": "WUA_STAT_25K_011361", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.015, sens=0.97, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.110.", "meta": {"id": "WUA_STAT_25K_011362", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.46, μ=18.3, σ=8.1, compute z."}]}, "response": "z = 0.267.", "meta": {"id": "WUA_STAT_25K_011363", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.48, s=36.6, n=53, compute CI."}]}, "response": "95% CI ≈ [53.63, 73.33].", "meta": {"id": "WUA_STAT_25K_011364", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.51, s=14.62, n=97, compute CI."}]}, "response": "95% CI ≈ [62.60, 68.42].", "meta": {"id": "WUA_STAT_25K_011365", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.16, compute E and Var."}]}, "response": "E[X]=4.80; Var[X]=4.032.", "meta": {"id": "WUA_STAT_25K_011366", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.56, s=22.65, n=122, compute CI."}]}, "response": "95% CI ≈ [79.54, 87.58].", "meta": {"id": "WUA_STAT_25K_011367", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.09, μ=45.25, σ=6.91, compute z."}]}, "response": "z = -2.773.", "meta": {"id": "WUA_STAT_25K_011368", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.043, sens=0.9, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.144.", "meta": {"id": "WUA_STAT_25K_011369", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.019, sens=0.87, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.115.", "meta": {"id": "WUA_STAT_25K_011370", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=183.25, s=16.89, n=88, compute CI."}]}, "response": "95% CI ≈ [179.72, 186.78].", "meta": {"id": "WUA_STAT_25K_011371", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.32, μ=79.45, σ=5.53, compute z."}]}, "response": "z = -2.193.", "meta": {"id": "WUA_STAT_25K_011372", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.197, sens=0.86, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.458.", "meta": {"id": "WUA_STAT_25K_011373", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.1, compute E and Var."}]}, "response": "E[X]=1.10; Var[X]=0.990.", "meta": {"id": "WUA_STAT_25K_011374", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.249, sens=0.82, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.502.", "meta": {"id": "WUA_STAT_25K_011375", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.5, compute E and Var."}]}, "response": "E[X]=7.00; Var[X]=3.500.", "meta": {"id": "WUA_STAT_25K_011376", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.97, μ=80.29, σ=3.7, compute z."}]}, "response": "z = -14.681.", "meta": {"id": "WUA_STAT_25K_011377", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.29, compute E and Var."}]}, "response": "E[X]=5.80; Var[X]=4.118.", "meta": {"id": "WUA_STAT_25K_011378", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.24, μ=90.96, σ=20.79, compute z."}]}, "response": "z = -3.257.", "meta": {"id": "WUA_STAT_25K_011379", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.16, compute E and Var."}]}, "response": "E[X]=4.64; Var[X]=3.898.", "meta": {"id": "WUA_STAT_25K_011380", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=68.57, s=9.86, n=161, compute CI."}]}, "response": "95% CI ≈ [67.05, 70.09].", "meta": {"id": "WUA_STAT_25K_011381", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.33, s=12.74, n=11, compute CI."}]}, "response": "95% CI ≈ [151.80, 166.86].", "meta": {"id": "WUA_STAT_25K_011382", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.0, μ=11.58, σ=2.01, compute z."}]}, "response": "z = 18.119.", "meta": {"id": "WUA_STAT_25K_011383", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=158.64, s=30.79, n=112, compute CI."}]}, "response": "95% CI ≈ [152.94, 164.34].", "meta": {"id": "WUA_STAT_25K_011384", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.56, s=28.99, n=21, compute CI."}]}, "response": "95% CI ≈ [0.16, 24.96].", "meta": {"id": "WUA_STAT_25K_011385", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.64, s=30.23, n=87, compute CI."}]}, "response": "95% CI ≈ [30.29, 42.99].", "meta": {"id": "WUA_STAT_25K_011386", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.05, sens=0.95, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.714.", "meta": {"id": "WUA_STAT_25K_011387", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.64, s=12.88, n=158, compute CI."}]}, "response": "95% CI ≈ [125.63, 129.65].", "meta": {"id": "WUA_STAT_25K_011388", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.87, s=11.38, n=148, compute CI."}]}, "response": "95% CI ≈ [73.04, 76.70].", "meta": {"id": "WUA_STAT_25K_011389", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.87, μ=79.15, σ=23.74, compute z."}]}, "response": "z = -0.559.", "meta": {"id": "WUA_STAT_25K_011390", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.79, s=26.14, n=167, compute CI."}]}, "response": "95% CI ≈ [7.83, 15.75].", "meta": {"id": "WUA_STAT_25K_011391", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.93, s=24.03, n=86, compute CI."}]}, "response": "95% CI ≈ [117.85, 128.01].", "meta": {"id": "WUA_STAT_25K_011392", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.56, μ=78.49, σ=1.54, compute z."}]}, "response": "z = -24.630.", "meta": {"id": "WUA_STAT_25K_011393", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.49, μ=40.46, σ=20.23, compute z."}]}, "response": "z = 2.275.", "meta": {"id": "WUA_STAT_25K_011394", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.77, compute E and Var."}]}, "response": "E[X]=21.56; Var[X]=4.959.", "meta": {"id": "WUA_STAT_25K_011395", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.3, μ=9.94, σ=1.15, compute z."}]}, "response": "z = 51.617.", "meta": {"id": "WUA_STAT_25K_011396", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.37, μ=68.54, σ=9.66, compute z."}]}, "response": "z = -1.984.", "meta": {"id": "WUA_STAT_25K_011397", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.93, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.481.", "meta": {"id": "WUA_STAT_25K_011398", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.81, s=30.37, n=126, compute CI."}]}, "response": "95% CI ≈ [145.51, 156.11].", "meta": {"id": "WUA_STAT_25K_011399", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.22, μ=24.54, σ=3.16, compute z."}]}, "response": "z = 6.861.", "meta": {"id": "WUA_STAT_25K_011400", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.042, sens=0.82, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.118.", "meta": {"id": "WUA_STAT_25K_011401", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.45, s=28.21, n=82, compute CI."}]}, "response": "95% CI ≈ [132.34, 144.56].", "meta": {"id": "WUA_STAT_25K_011402", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.83, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.536.", "meta": {"id": "WUA_STAT_25K_011403", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.087, sens=0.8, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.718.", "meta": {"id": "WUA_STAT_25K_011404", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.049, sens=0.72, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.171.", "meta": {"id": "WUA_STAT_25K_011405", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.67, compute E and Var."}]}, "response": "E[X]=9.38; Var[X]=3.095.", "meta": {"id": "WUA_STAT_25K_011406", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=96.69, s=32.55, n=110, compute CI."}]}, "response": "95% CI ≈ [90.61, 102.77].", "meta": {"id": "WUA_STAT_25K_011407", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.74, compute E and Var."}]}, "response": "E[X]=26.64; Var[X]=6.926.", "meta": {"id": "WUA_STAT_25K_011408", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.73, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.506.", "meta": {"id": "WUA_STAT_25K_011409", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.76, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.619.", "meta": {"id": "WUA_STAT_25K_011410", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.45, s=22.32, n=165, compute CI."}]}, "response": "95% CI ≈ [186.04, 192.86].", "meta": {"id": "WUA_STAT_25K_011411", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.57, compute E and Var."}]}, "response": "E[X]=17.67; Var[X]=7.598.", "meta": {"id": "WUA_STAT_25K_011412", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.1, s=28.21, n=16, compute CI."}]}, "response": "95% CI ≈ [109.28, 136.92].", "meta": {"id": "WUA_STAT_25K_011413", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.82, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.697.", "meta": {"id": "WUA_STAT_25K_011414", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.13, compute E and Var."}]}, "response": "E[X]=0.91; Var[X]=0.792.", "meta": {"id": "WUA_STAT_25K_011415", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.209, sens=0.81, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.754.", "meta": {"id": "WUA_STAT_25K_011416", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.3, compute E and Var."}]}, "response": "E[X]=6.60; Var[X]=4.620.", "meta": {"id": "WUA_STAT_25K_011417", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=68.52, s=10.83, n=123, compute CI."}]}, "response": "95% CI ≈ [66.61, 70.43].", "meta": {"id": "WUA_STAT_25K_011418", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.096, sens=0.98, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.321.", "meta": {"id": "WUA_STAT_25K_011419", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.78, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.525.", "meta": {"id": "WUA_STAT_25K_011420", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.37, compute E and Var."}]}, "response": "E[X]=7.77; Var[X]=4.895.", "meta": {"id": "WUA_STAT_25K_011421", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.71, compute E and Var."}]}, "response": "E[X]=17.75; Var[X]=5.148.", "meta": {"id": "WUA_STAT_25K_011422", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.128, sens=0.98, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.878.", "meta": {"id": "WUA_STAT_25K_011423", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.14, compute E and Var."}]}, "response": "E[X]=2.38; Var[X]=2.047.", "meta": {"id": "WUA_STAT_25K_011424", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.105, sens=0.74, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.237.", "meta": {"id": "WUA_STAT_25K_011425", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=158.11, s=3.92, n=72, compute CI."}]}, "response": "95% CI ≈ [157.20, 159.02].", "meta": {"id": "WUA_STAT_25K_011426", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.76, s=21.85, n=105, compute CI."}]}, "response": "95% CI ≈ [148.58, 156.94].", "meta": {"id": "WUA_STAT_25K_011427", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.15, compute E and Var."}]}, "response": "E[X]=1.20; Var[X]=1.020.", "meta": {"id": "WUA_STAT_25K_011428", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.8, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.675.", "meta": {"id": "WUA_STAT_25K_011429", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.32, μ=55.05, σ=19.8, compute z."}]}, "response": "z = -0.946.", "meta": {"id": "WUA_STAT_25K_011430", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.2, μ=8.84, σ=24.1, compute z."}]}, "response": "z = 3.583.", "meta": {"id": "WUA_STAT_25K_011431", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.76, μ=48.13, σ=2.26, compute z."}]}, "response": "z = -17.863.", "meta": {"id": "WUA_STAT_25K_011432", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.48, s=23.31, n=129, compute CI."}]}, "response": "95% CI ≈ [45.46, 53.50].", "meta": {"id": "WUA_STAT_25K_011433", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.64, μ=0.35, σ=6.07, compute z."}]}, "response": "z = 2.025.", "meta": {"id": "WUA_STAT_25K_011434", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.57, compute E and Var."}]}, "response": "E[X]=11.40; Var[X]=4.902.", "meta": {"id": "WUA_STAT_25K_011435", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.42, s=30.0, n=126, compute CI."}]}, "response": "95% CI ≈ [176.18, 186.66].", "meta": {"id": "WUA_STAT_25K_011436", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.35, compute E and Var."}]}, "response": "E[X]=7.00; Var[X]=4.550.", "meta": {"id": "WUA_STAT_25K_011437", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.19, μ=33.21, σ=1.66, compute z."}]}, "response": "z = -1.217.", "meta": {"id": "WUA_STAT_25K_011438", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.213, sens=0.92, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.892.", "meta": {"id": "WUA_STAT_25K_011439", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.98, s=7.3, n=199, compute CI."}]}, "response": "95% CI ≈ [173.97, 175.99].", "meta": {"id": "WUA_STAT_25K_011440", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.15, s=3.04, n=18, compute CI."}]}, "response": "95% CI ≈ [185.75, 188.55].", "meta": {"id": "WUA_STAT_25K_011441", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.15, compute E and Var."}]}, "response": "E[X]=2.25; Var[X]=1.912.", "meta": {"id": "WUA_STAT_25K_011442", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.9, μ=67.31, σ=9.89, compute z."}]}, "response": "z = -0.547.", "meta": {"id": "WUA_STAT_25K_011443", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.91, s=32.93, n=115, compute CI."}]}, "response": "95% CI ≈ [148.89, 160.93].", "meta": {"id": "WUA_STAT_25K_011444", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.97, s=19.89, n=167, compute CI."}]}, "response": "95% CI ≈ [190.95, 196.99].", "meta": {"id": "WUA_STAT_25K_011445", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.24, compute E and Var."}]}, "response": "E[X]=6.48; Var[X]=4.925.", "meta": {"id": "WUA_STAT_25K_011446", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.063, sens=0.9, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.302.", "meta": {"id": "WUA_STAT_25K_011447", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.19, compute E and Var."}]}, "response": "E[X]=6.65; Var[X]=5.387.", "meta": {"id": "WUA_STAT_25K_011448", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.4, μ=2.34, σ=11.51, compute z."}]}, "response": "z = 5.218.", "meta": {"id": "WUA_STAT_25K_011449", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.82, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.622.", "meta": {"id": "WUA_STAT_25K_011450", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.9, s=8.85, n=101, compute CI."}]}, "response": "95% CI ≈ [180.17, 183.63].", "meta": {"id": "WUA_STAT_25K_011451", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.2, μ=26.7, σ=9.8, compute z."}]}, "response": "z = 5.357.", "meta": {"id": "WUA_STAT_25K_011452", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.39, s=24.3, n=92, compute CI."}]}, "response": "95% CI ≈ [22.42, 32.36].", "meta": {"id": "WUA_STAT_25K_011453", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.063, sens=0.71, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.323.", "meta": {"id": "WUA_STAT_25K_011454", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.72, s=27.31, n=185, compute CI."}]}, "response": "95% CI ≈ [94.78, 102.66].", "meta": {"id": "WUA_STAT_25K_011455", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.73, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.243.", "meta": {"id": "WUA_STAT_25K_011456", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.195, sens=0.84, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.504.", "meta": {"id": "WUA_STAT_25K_011457", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.15, compute E and Var."}]}, "response": "E[X]=2.85; Var[X]=2.422.", "meta": {"id": "WUA_STAT_25K_011458", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.74, μ=93.5, σ=20.72, compute z."}]}, "response": "z = 0.108.", "meta": {"id": "WUA_STAT_25K_011459", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.64, compute E and Var."}]}, "response": "E[X]=20.48; Var[X]=7.373.", "meta": {"id": "WUA_STAT_25K_011460", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.015, sens=0.77, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.042.", "meta": {"id": "WUA_STAT_25K_011461", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.85, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.748.", "meta": {"id": "WUA_STAT_25K_011462", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.82, μ=36.64, σ=10.82, compute z."}]}, "response": "z = 2.789.", "meta": {"id": "WUA_STAT_25K_011463", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.099, sens=0.94, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.378.", "meta": {"id": "WUA_STAT_25K_011464", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.41, μ=8.53, σ=18.87, compute z."}]}, "response": "z = 3.279.", "meta": {"id": "WUA_STAT_25K_011465", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.36, μ=92.64, σ=6.12, compute z."}]}, "response": "z = -4.621.", "meta": {"id": "WUA_STAT_25K_011466", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.91, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.712.", "meta": {"id": "WUA_STAT_25K_011467", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.66, compute E and Var."}]}, "response": "E[X]=13.20; Var[X]=4.488.", "meta": {"id": "WUA_STAT_25K_011468", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.05, s=3.73, n=110, compute CI."}]}, "response": "95% CI ≈ [86.35, 87.75].", "meta": {"id": "WUA_STAT_25K_011469", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.1, μ=37.47, σ=8.2, compute z."}]}, "response": "z = 1.174.", "meta": {"id": "WUA_STAT_25K_011470", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.75, μ=76.98, σ=2.99, compute z."}]}, "response": "z = -12.117.", "meta": {"id": "WUA_STAT_25K_011471", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.86, s=2.2, n=63, compute CI."}]}, "response": "95% CI ≈ [22.32, 23.40].", "meta": {"id": "WUA_STAT_25K_011472", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.51, s=5.94, n=88, compute CI."}]}, "response": "95% CI ≈ [62.27, 64.75].", "meta": {"id": "WUA_STAT_25K_011473", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.77, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.680.", "meta": {"id": "WUA_STAT_25K_011474", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.86, s=35.88, n=118, compute CI."}]}, "response": "95% CI ≈ [147.39, 160.33].", "meta": {"id": "WUA_STAT_25K_011475", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.92, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.440.", "meta": {"id": "WUA_STAT_25K_011476", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.123, sens=0.87, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.357.", "meta": {"id": "WUA_STAT_25K_011477", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.79, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.571.", "meta": {"id": "WUA_STAT_25K_011478", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.89, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.756.", "meta": {"id": "WUA_STAT_25K_011479", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.76, compute E and Var."}]}, "response": "E[X]=27.36; Var[X]=6.566.", "meta": {"id": "WUA_STAT_25K_011480", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.62, compute E and Var."}]}, "response": "E[X]=13.64; Var[X]=5.183.", "meta": {"id": "WUA_STAT_25K_011481", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.8, compute E and Var."}]}, "response": "E[X]=12.00; Var[X]=2.400.", "meta": {"id": "WUA_STAT_25K_011482", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.14, compute E and Var."}]}, "response": "E[X]=1.12; Var[X]=0.963.", "meta": {"id": "WUA_STAT_25K_011483", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.093, sens=0.83, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.378.", "meta": {"id": "WUA_STAT_25K_011484", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.93, s=18.82, n=14, compute CI."}]}, "response": "95% CI ≈ [65.07, 84.79].", "meta": {"id": "WUA_STAT_25K_011485", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.09, compute E and Var."}]}, "response": "E[X]=0.72; Var[X]=0.655.", "meta": {"id": "WUA_STAT_25K_011486", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.01, s=16.31, n=131, compute CI."}]}, "response": "95% CI ≈ [104.22, 109.80].", "meta": {"id": "WUA_STAT_25K_011487", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.74, compute E and Var."}]}, "response": "E[X]=20.72; Var[X]=5.387.", "meta": {"id": "WUA_STAT_25K_011488", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.89, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.785.", "meta": {"id": "WUA_STAT_25K_011489", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.24, compute E and Var."}]}, "response": "E[X]=8.16; Var[X]=6.202.", "meta": {"id": "WUA_STAT_25K_011490", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.61, compute E and Var."}]}, "response": "E[X]=17.08; Var[X]=6.661.", "meta": {"id": "WUA_STAT_25K_011491", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.39, μ=47.28, σ=12.94, compute z."}]}, "response": "z = -1.228.", "meta": {"id": "WUA_STAT_25K_011492", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.066, sens=0.73, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.462.", "meta": {"id": "WUA_STAT_25K_011493", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.32, μ=43.24, σ=3.3, compute z."}]}, "response": "z = -1.794.", "meta": {"id": "WUA_STAT_25K_011494", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.14, compute E and Var."}]}, "response": "E[X]=3.92; Var[X]=3.371.", "meta": {"id": "WUA_STAT_25K_011495", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.7, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.295.", "meta": {"id": "WUA_STAT_25K_011496", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.53, μ=46.0, σ=6.55, compute z."}]}, "response": "z = 6.340.", "meta": {"id": "WUA_STAT_25K_011497", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.85, μ=38.24, σ=19.6, compute z."}]}, "response": "z = 0.286.", "meta": {"id": "WUA_STAT_25K_011498", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.07, μ=94.48, σ=15.07, compute z."}]}, "response": "z = -5.601.", "meta": {"id": "WUA_STAT_25K_011499", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.23, compute E and Var."}]}, "response": "E[X]=1.15; Var[X]=0.886.", "meta": {"id": "WUA_STAT_25K_011500", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.06, compute E and Var."}]}, "response": "E[X]=0.42; Var[X]=0.395.", "meta": {"id": "WUA_STAT_25K_011501", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.31, compute E and Var."}]}, "response": "E[X]=8.68; Var[X]=5.989.", "meta": {"id": "WUA_STAT_25K_011502", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.84, μ=56.08, σ=6.61, compute z."}]}, "response": "z = -8.054.", "meta": {"id": "WUA_STAT_25K_011503", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.82, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.656.", "meta": {"id": "WUA_STAT_25K_011504", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.26, μ=77.21, σ=10.19, compute z."}]}, "response": "z = -3.822.", "meta": {"id": "WUA_STAT_25K_011505", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.21, s=35.82, n=170, compute CI."}]}, "response": "95% CI ≈ [140.83, 151.59].", "meta": {"id": "WUA_STAT_25K_011506", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.71, s=19.18, n=106, compute CI."}]}, "response": "95% CI ≈ [147.06, 154.36].", "meta": {"id": "WUA_STAT_25K_011507", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.79, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.304.", "meta": {"id": "WUA_STAT_25K_011508", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.14, compute E and Var."}]}, "response": "E[X]=3.50; Var[X]=3.010.", "meta": {"id": "WUA_STAT_25K_011509", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.9, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.475.", "meta": {"id": "WUA_STAT_25K_011510", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.96, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.455.", "meta": {"id": "WUA_STAT_25K_011511", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.091, sens=0.84, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.584.", "meta": {"id": "WUA_STAT_25K_011512", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.7, s=13.73, n=29, compute CI."}]}, "response": "95% CI ≈ [104.70, 114.70].", "meta": {"id": "WUA_STAT_25K_011513", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.052, sens=0.7, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.228.", "meta": {"id": "WUA_STAT_25K_011514", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.13, μ=88.84, σ=12.55, compute z."}]}, "response": "z = -6.272.", "meta": {"id": "WUA_STAT_25K_011515", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.47, μ=95.28, σ=11.99, compute z."}]}, "response": "z = -5.656.", "meta": {"id": "WUA_STAT_25K_011516", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.122, sens=0.84, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.539.", "meta": {"id": "WUA_STAT_25K_011517", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.66, compute E and Var."}]}, "response": "E[X]=23.10; Var[X]=7.854.", "meta": {"id": "WUA_STAT_25K_011518", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.117, sens=0.94, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.384.", "meta": {"id": "WUA_STAT_25K_011519", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.085, sens=0.71, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.231.", "meta": {"id": "WUA_STAT_25K_011520", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.83, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.539.", "meta": {"id": "WUA_STAT_25K_011521", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.149, sens=0.71, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.332.", "meta": {"id": "WUA_STAT_25K_011522", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.32, s=35.21, n=18, compute CI."}]}, "response": "95% CI ≈ [154.05, 186.59].", "meta": {"id": "WUA_STAT_25K_011523", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.92, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.337.", "meta": {"id": "WUA_STAT_25K_011524", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.21, compute E and Var."}]}, "response": "E[X]=3.78; Var[X]=2.986.", "meta": {"id": "WUA_STAT_25K_011525", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=94.08, s=33.69, n=90, compute CI."}]}, "response": "95% CI ≈ [87.12, 101.04].", "meta": {"id": "WUA_STAT_25K_011526", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.244, sens=0.76, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.654.", "meta": {"id": "WUA_STAT_25K_011527", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.49, compute E and Var."}]}, "response": "E[X]=6.37; Var[X]=3.249.", "meta": {"id": "WUA_STAT_25K_011528", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.9, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.362.", "meta": {"id": "WUA_STAT_25K_011529", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.33, s=24.09, n=89, compute CI."}]}, "response": "95% CI ≈ [39.33, 49.33].", "meta": {"id": "WUA_STAT_25K_011530", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.54, s=6.18, n=124, compute CI."}]}, "response": "95% CI ≈ [14.45, 16.63].", "meta": {"id": "WUA_STAT_25K_011531", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.49, compute E and Var."}]}, "response": "E[X]=10.78; Var[X]=5.498.", "meta": {"id": "WUA_STAT_25K_011532", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.22, compute E and Var."}]}, "response": "E[X]=6.38; Var[X]=4.976.", "meta": {"id": "WUA_STAT_25K_011533", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.02, μ=5.28, σ=8.27, compute z."}]}, "response": "z = 1.903.", "meta": {"id": "WUA_STAT_25K_011534", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=51.98, s=21.11, n=70, compute CI."}]}, "response": "95% CI ≈ [47.03, 56.93].", "meta": {"id": "WUA_STAT_25K_011535", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.53, μ=47.01, σ=15.15, compute z."}]}, "response": "z = -2.078.", "meta": {"id": "WUA_STAT_25K_011536", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.118, sens=0.87, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.538.", "meta": {"id": "WUA_STAT_25K_011537", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.7, compute E and Var."}]}, "response": "E[X]=6.30; Var[X]=1.890.", "meta": {"id": "WUA_STAT_25K_011538", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.15, compute E and Var."}]}, "response": "E[X]=2.70; Var[X]=2.295.", "meta": {"id": "WUA_STAT_25K_011539", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.1, s=4.68, n=14, compute CI."}]}, "response": "95% CI ≈ [9.65, 14.55].", "meta": {"id": "WUA_STAT_25K_011540", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.44, μ=56.95, σ=21.04, compute z."}]}, "response": "z = -0.357.", "meta": {"id": "WUA_STAT_25K_011541", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.28, μ=63.82, σ=11.01, compute z."}]}, "response": "z = -3.500.", "meta": {"id": "WUA_STAT_25K_011542", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.43, compute E and Var."}]}, "response": "E[X]=5.16; Var[X]=2.941.", "meta": {"id": "WUA_STAT_25K_011543", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.247, sens=0.92, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.699.", "meta": {"id": "WUA_STAT_25K_011544", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.3, s=9.68, n=95, compute CI."}]}, "response": "95% CI ≈ [152.35, 156.25].", "meta": {"id": "WUA_STAT_25K_011545", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.14, μ=3.09, σ=1.05, compute z."}]}, "response": "z = 29.571.", "meta": {"id": "WUA_STAT_25K_011546", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=71.05, s=11.01, n=85, compute CI."}]}, "response": "95% CI ≈ [68.71, 73.39].", "meta": {"id": "WUA_STAT_25K_011547", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=155.6, s=12.29, n=40, compute CI."}]}, "response": "95% CI ≈ [151.79, 159.41].", "meta": {"id": "WUA_STAT_25K_011548", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.73, μ=39.3, σ=7.74, compute z."}]}, "response": "z = -1.366.", "meta": {"id": "WUA_STAT_25K_011549", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.167, sens=0.88, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.434.", "meta": {"id": "WUA_STAT_25K_011550", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.146, sens=0.74, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.559.", "meta": {"id": "WUA_STAT_25K_011551", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.73, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.484.", "meta": {"id": "WUA_STAT_25K_011552", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.73, μ=20.89, σ=13.63, compute z."}]}, "response": "z = 3.877.", "meta": {"id": "WUA_STAT_25K_011553", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.19, μ=98.87, σ=20.48, compute z."}]}, "response": "z = -0.814.", "meta": {"id": "WUA_STAT_25K_011554", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.39, compute E and Var."}]}, "response": "E[X]=15.60; Var[X]=9.516.", "meta": {"id": "WUA_STAT_25K_011555", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.44, s=12.54, n=175, compute CI."}]}, "response": "95% CI ≈ [182.58, 186.30].", "meta": {"id": "WUA_STAT_25K_011556", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.1, μ=5.87, σ=8.72, compute z."}]}, "response": "z = 2.091.", "meta": {"id": "WUA_STAT_25K_011557", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.76, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.482.", "meta": {"id": "WUA_STAT_25K_011558", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.75, s=24.0, n=193, compute CI."}]}, "response": "95% CI ≈ [154.36, 161.14].", "meta": {"id": "WUA_STAT_25K_011559", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.63, s=27.69, n=38, compute CI."}]}, "response": "95% CI ≈ [49.83, 67.43].", "meta": {"id": "WUA_STAT_25K_011560", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.68, compute E and Var."}]}, "response": "E[X]=14.96; Var[X]=4.787.", "meta": {"id": "WUA_STAT_25K_011561", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.62, μ=44.28, σ=1.99, compute z."}]}, "response": "z = -9.879.", "meta": {"id": "WUA_STAT_25K_011562", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.2, s=37.58, n=166, compute CI."}]}, "response": "95% CI ≈ [103.48, 114.92].", "meta": {"id": "WUA_STAT_25K_011563", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.43, s=12.25, n=29, compute CI."}]}, "response": "95% CI ≈ [64.97, 73.89].", "meta": {"id": "WUA_STAT_25K_011564", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.25, μ=15.35, σ=22.71, compute z."}]}, "response": "z = -0.092.", "meta": {"id": "WUA_STAT_25K_011565", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=147.39, s=22.0, n=123, compute CI."}]}, "response": "95% CI ≈ [143.50, 151.28].", "meta": {"id": "WUA_STAT_25K_011566", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.38, μ=62.02, σ=15.4, compute z."}]}, "response": "z = -3.743.", "meta": {"id": "WUA_STAT_25K_011567", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.14, μ=72.89, σ=11.48, compute z."}]}, "response": "z = -5.030.", "meta": {"id": "WUA_STAT_25K_011568", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.2, sens=0.71, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.617.", "meta": {"id": "WUA_STAT_25K_011569", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.048, sens=0.72, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.136.", "meta": {"id": "WUA_STAT_25K_011570", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.087, sens=0.98, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.272.", "meta": {"id": "WUA_STAT_25K_011571", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.034, sens=0.75, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.086.", "meta": {"id": "WUA_STAT_25K_011572", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=47.11, s=33.96, n=189, compute CI."}]}, "response": "95% CI ≈ [42.27, 51.95].", "meta": {"id": "WUA_STAT_25K_011573", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.01, μ=96.15, σ=7.76, compute z."}]}, "response": "z = -6.848.", "meta": {"id": "WUA_STAT_25K_011574", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.37, compute E and Var."}]}, "response": "E[X]=12.58; Var[X]=7.925.", "meta": {"id": "WUA_STAT_25K_011575", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=51.2, s=21.42, n=34, compute CI."}]}, "response": "95% CI ≈ [44.00, 58.40].", "meta": {"id": "WUA_STAT_25K_011576", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.69, s=26.54, n=71, compute CI."}]}, "response": "95% CI ≈ [150.52, 162.86].", "meta": {"id": "WUA_STAT_25K_011577", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.22, compute E and Var."}]}, "response": "E[X]=4.84; Var[X]=3.775.", "meta": {"id": "WUA_STAT_25K_011578", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.78, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.581.", "meta": {"id": "WUA_STAT_25K_011579", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.77, s=5.83, n=153, compute CI."}]}, "response": "95% CI ≈ [53.85, 55.69].", "meta": {"id": "WUA_STAT_25K_011580", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.71, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.385.", "meta": {"id": "WUA_STAT_25K_011581", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.35, μ=67.13, σ=9.37, compute z."}]}, "response": "z = 1.091.", "meta": {"id": "WUA_STAT_25K_011582", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.179, sens=0.78, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.531.", "meta": {"id": "WUA_STAT_25K_011583", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.71, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.841.", "meta": {"id": "WUA_STAT_25K_011584", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.89, s=16.59, n=34, compute CI."}]}, "response": "95% CI ≈ [65.31, 76.47].", "meta": {"id": "WUA_STAT_25K_011585", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.055, sens=0.88, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.561.", "meta": {"id": "WUA_STAT_25K_011586", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.98, s=14.34, n=184, compute CI."}]}, "response": "95% CI ≈ [72.91, 77.05].", "meta": {"id": "WUA_STAT_25K_011587", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.9, s=14.69, n=191, compute CI."}]}, "response": "95% CI ≈ [44.82, 48.98].", "meta": {"id": "WUA_STAT_25K_011588", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.237, sens=0.71, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.565.", "meta": {"id": "WUA_STAT_25K_011589", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.69, compute E and Var."}]}, "response": "E[X]=12.42; Var[X]=3.850.", "meta": {"id": "WUA_STAT_25K_011590", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.48, compute E and Var."}]}, "response": "E[X]=12.96; Var[X]=6.739.", "meta": {"id": "WUA_STAT_25K_011591", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.6, μ=64.47, σ=23.21, compute z."}]}, "response": "z = -0.425.", "meta": {"id": "WUA_STAT_25K_011592", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.19, μ=44.51, σ=22.47, compute z."}]}, "response": "z = 1.454.", "meta": {"id": "WUA_STAT_25K_011593", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.79, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.494.", "meta": {"id": "WUA_STAT_25K_011594", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.58, μ=51.56, σ=4.46, compute z."}]}, "response": "z = -3.359.", "meta": {"id": "WUA_STAT_25K_011595", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.2, sens=0.9, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.738.", "meta": {"id": "WUA_STAT_25K_011596", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.92, μ=40.23, σ=15.17, compute z."}]}, "response": "z = 2.616.", "meta": {"id": "WUA_STAT_25K_011597", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.28, s=39.79, n=91, compute CI."}]}, "response": "95% CI ≈ [177.10, 193.46].", "meta": {"id": "WUA_STAT_25K_011598", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.06, μ=70.14, σ=17.73, compute z."}]}, "response": "z = -1.302.", "meta": {"id": "WUA_STAT_25K_011599", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.47, μ=36.0, σ=8.56, compute z."}]}, "response": "z = 2.625.", "meta": {"id": "WUA_STAT_25K_011600", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.94, μ=66.45, σ=23.23, compute z."}]}, "response": "z = -0.668.", "meta": {"id": "WUA_STAT_25K_011601", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.033, sens=0.74, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.457.", "meta": {"id": "WUA_STAT_25K_011602", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.62, compute E and Var."}]}, "response": "E[X]=7.44; Var[X]=2.827.", "meta": {"id": "WUA_STAT_25K_011603", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.73, s=30.86, n=119, compute CI."}]}, "response": "95% CI ≈ [176.19, 187.27].", "meta": {"id": "WUA_STAT_25K_011604", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.21, s=14.88, n=79, compute CI."}]}, "response": "95% CI ≈ [147.93, 154.49].", "meta": {"id": "WUA_STAT_25K_011605", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.54, compute E and Var."}]}, "response": "E[X]=12.96; Var[X]=5.962.", "meta": {"id": "WUA_STAT_25K_011606", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.98, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.484.", "meta": {"id": "WUA_STAT_25K_011607", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.93, s=26.06, n=160, compute CI."}]}, "response": "95% CI ≈ [32.89, 40.97].", "meta": {"id": "WUA_STAT_25K_011608", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.71, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.673.", "meta": {"id": "WUA_STAT_25K_011609", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.68, s=16.2, n=180, compute CI."}]}, "response": "95% CI ≈ [83.31, 88.05].", "meta": {"id": "WUA_STAT_25K_011610", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.72, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.220.", "meta": {"id": "WUA_STAT_25K_011611", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.14, μ=84.47, σ=2.13, compute z."}]}, "response": "z = 1.723.", "meta": {"id": "WUA_STAT_25K_011612", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.32, compute E and Var."}]}, "response": "E[X]=8.32; Var[X]=5.658.", "meta": {"id": "WUA_STAT_25K_011613", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.63, compute E and Var."}]}, "response": "E[X]=7.56; Var[X]=2.797.", "meta": {"id": "WUA_STAT_25K_011614", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.59, μ=26.06, σ=11.05, compute z."}]}, "response": "z = 5.478.", "meta": {"id": "WUA_STAT_25K_011615", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.44, compute E and Var."}]}, "response": "E[X]=10.12; Var[X]=5.667.", "meta": {"id": "WUA_STAT_25K_011616", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.28, μ=6.47, σ=19.85, compute z."}]}, "response": "z = 2.963.", "meta": {"id": "WUA_STAT_25K_011617", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.08, μ=44.91, σ=16.64, compute z."}]}, "response": "z = 0.251.", "meta": {"id": "WUA_STAT_25K_011618", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.157, sens=0.97, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.819.", "meta": {"id": "WUA_STAT_25K_011619", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.12, μ=20.31, σ=2.56, compute z."}]}, "response": "z = 28.832.", "meta": {"id": "WUA_STAT_25K_011620", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.047, sens=0.89, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.814.", "meta": {"id": "WUA_STAT_25K_011621", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.47, compute E and Var."}]}, "response": "E[X]=16.45; Var[X]=8.719.", "meta": {"id": "WUA_STAT_25K_011622", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.07, μ=38.47, σ=19.94, compute z."}]}, "response": "z = -0.672.", "meta": {"id": "WUA_STAT_25K_011623", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.84, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.922.", "meta": {"id": "WUA_STAT_25K_011624", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.058, sens=0.86, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.159.", "meta": {"id": "WUA_STAT_25K_011625", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.26, s=30.4, n=49, compute CI."}]}, "response": "95% CI ≈ [137.75, 154.77].", "meta": {"id": "WUA_STAT_25K_011626", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.16, μ=93.71, σ=15.88, compute z."}]}, "response": "z = 0.154.", "meta": {"id": "WUA_STAT_25K_011627", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.71, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.920.", "meta": {"id": "WUA_STAT_25K_011628", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.92, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.340.", "meta": {"id": "WUA_STAT_25K_011629", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.192, sens=0.87, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.697.", "meta": {"id": "WUA_STAT_25K_011630", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.88, μ=81.63, σ=9.44, compute z."}]}, "response": "z = -2.834.", "meta": {"id": "WUA_STAT_25K_011631", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.77, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.733.", "meta": {"id": "WUA_STAT_25K_011632", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.5, s=30.63, n=160, compute CI."}]}, "response": "95% CI ≈ [112.75, 122.25].", "meta": {"id": "WUA_STAT_25K_011633", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.7, compute E and Var."}]}, "response": "E[X]=28.00; Var[X]=8.400.", "meta": {"id": "WUA_STAT_25K_011634", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.29, compute E and Var."}]}, "response": "E[X]=11.60; Var[X]=8.236.", "meta": {"id": "WUA_STAT_25K_011635", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.093, sens=0.83, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.334.", "meta": {"id": "WUA_STAT_25K_011636", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=64.03, s=30.38, n=81, compute CI."}]}, "response": "95% CI ≈ [57.41, 70.65].", "meta": {"id": "WUA_STAT_25K_011637", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=191.99, s=27.87, n=171, compute CI."}]}, "response": "95% CI ≈ [187.81, 196.17].", "meta": {"id": "WUA_STAT_25K_011638", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.49, compute E and Var."}]}, "response": "E[X]=11.27; Var[X]=5.748.", "meta": {"id": "WUA_STAT_25K_011639", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.15, μ=63.23, σ=6.78, compute z."}]}, "response": "z = -3.699.", "meta": {"id": "WUA_STAT_25K_011640", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.8, compute E and Var."}]}, "response": "E[X]=9.60; Var[X]=1.920.", "meta": {"id": "WUA_STAT_25K_011641", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.32, μ=88.27, σ=23.75, compute z."}]}, "response": "z = -1.345.", "meta": {"id": "WUA_STAT_25K_011642", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.6, s=36.86, n=14, compute CI."}]}, "response": "95% CI ≈ [17.29, 55.91].", "meta": {"id": "WUA_STAT_25K_011643", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.92, μ=15.16, σ=18.76, compute z."}]}, "response": "z = 2.226.", "meta": {"id": "WUA_STAT_25K_011644", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.89, μ=40.87, σ=20.13, compute z."}]}, "response": "z = -0.993.", "meta": {"id": "WUA_STAT_25K_011645", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.03, sens=0.74, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.081.", "meta": {"id": "WUA_STAT_25K_011646", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.3, s=11.13, n=180, compute CI."}]}, "response": "95% CI ≈ [27.67, 30.93].", "meta": {"id": "WUA_STAT_25K_011647", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.46, compute E and Var."}]}, "response": "E[X]=11.04; Var[X]=5.962.", "meta": {"id": "WUA_STAT_25K_011648", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.94, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.813.", "meta": {"id": "WUA_STAT_25K_011649", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.12, s=24.12, n=132, compute CI."}]}, "response": "95% CI ≈ [138.01, 146.23].", "meta": {"id": "WUA_STAT_25K_011650", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.95, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.111.", "meta": {"id": "WUA_STAT_25K_011651", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.64, compute E and Var."}]}, "response": "E[X]=3.84; Var[X]=1.382.", "meta": {"id": "WUA_STAT_25K_011652", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.24, compute E and Var."}]}, "response": "E[X]=8.88; Var[X]=6.749.", "meta": {"id": "WUA_STAT_25K_011653", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.18, sens=0.91, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.588.", "meta": {"id": "WUA_STAT_25K_011654", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.09, compute E and Var."}]}, "response": "E[X]=2.88; Var[X]=2.621.", "meta": {"id": "WUA_STAT_25K_011655", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.07, μ=19.04, σ=9.52, compute z."}]}, "response": "z = 5.991.", "meta": {"id": "WUA_STAT_25K_011656", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.36, μ=18.45, σ=1.34, compute z."}]}, "response": "z = -12.007.", "meta": {"id": "WUA_STAT_25K_011657", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.19, μ=66.51, σ=14.47, compute z."}]}, "response": "z = -3.616.", "meta": {"id": "WUA_STAT_25K_011658", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.2, compute E and Var."}]}, "response": "E[X]=2.80; Var[X]=2.240.", "meta": {"id": "WUA_STAT_25K_011659", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.84, μ=38.16, σ=22.23, compute z."}]}, "response": "z = 0.480.", "meta": {"id": "WUA_STAT_25K_011660", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.02, s=23.67, n=151, compute CI."}]}, "response": "95% CI ≈ [130.24, 137.80].", "meta": {"id": "WUA_STAT_25K_011661", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.168, sens=0.75, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.752.", "meta": {"id": "WUA_STAT_25K_011662", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.18, s=12.32, n=56, compute CI."}]}, "response": "95% CI ≈ [66.95, 73.41].", "meta": {"id": "WUA_STAT_25K_011663", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.55, s=23.74, n=198, compute CI."}]}, "response": "95% CI ≈ [33.24, 39.86].", "meta": {"id": "WUA_STAT_25K_011664", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=93.75, s=22.53, n=138, compute CI."}]}, "response": "95% CI ≈ [89.99, 97.51].", "meta": {"id": "WUA_STAT_25K_011665", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.29, μ=21.81, σ=9.78, compute z."}]}, "response": "z = 0.356.", "meta": {"id": "WUA_STAT_25K_011666", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.72, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.460.", "meta": {"id": "WUA_STAT_25K_011667", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.37, μ=22.31, σ=15.09, compute z."}]}, "response": "z = 0.600.", "meta": {"id": "WUA_STAT_25K_011668", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.94, s=13.95, n=74, compute CI."}]}, "response": "95% CI ≈ [181.76, 188.12].", "meta": {"id": "WUA_STAT_25K_011669", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.95, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.610.", "meta": {"id": "WUA_STAT_25K_011670", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=66.06, s=26.58, n=43, compute CI."}]}, "response": "95% CI ≈ [58.12, 74.00].", "meta": {"id": "WUA_STAT_25K_011671", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.7, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.806.", "meta": {"id": "WUA_STAT_25K_011672", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.71, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.487.", "meta": {"id": "WUA_STAT_25K_011673", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.17, sens=0.81, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.480.", "meta": {"id": "WUA_STAT_25K_011674", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.43, compute E and Var."}]}, "response": "E[X]=3.01; Var[X]=1.716.", "meta": {"id": "WUA_STAT_25K_011675", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.9, μ=77.11, σ=21.6, compute z."}]}, "response": "z = -1.584.", "meta": {"id": "WUA_STAT_25K_011676", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.27, s=1.0, n=185, compute CI."}]}, "response": "95% CI ≈ [38.13, 38.41].", "meta": {"id": "WUA_STAT_25K_011677", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.25, μ=88.82, σ=15.82, compute z."}]}, "response": "z = -2.564.", "meta": {"id": "WUA_STAT_25K_011678", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.65, compute E and Var."}]}, "response": "E[X]=24.70; Var[X]=8.645.", "meta": {"id": "WUA_STAT_25K_011679", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.83, s=20.67, n=43, compute CI."}]}, "response": "95% CI ≈ [21.65, 34.01].", "meta": {"id": "WUA_STAT_25K_011680", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.3, s=7.13, n=25, compute CI."}]}, "response": "95% CI ≈ [104.51, 110.09].", "meta": {"id": "WUA_STAT_25K_011681", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.68, compute E and Var."}]}, "response": "E[X]=10.20; Var[X]=3.264.", "meta": {"id": "WUA_STAT_25K_011682", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.27, μ=10.3, σ=8.93, compute z."}]}, "response": "z = 9.291.", "meta": {"id": "WUA_STAT_25K_011683", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.8, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.108.", "meta": {"id": "WUA_STAT_25K_011684", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.095, sens=0.75, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.226.", "meta": {"id": "WUA_STAT_25K_011685", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.0, μ=4.56, σ=3.84, compute z."}]}, "response": "z = 2.719.", "meta": {"id": "WUA_STAT_25K_011686", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.71, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.388.", "meta": {"id": "WUA_STAT_25K_011687", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.32, μ=4.68, σ=8.25, compute z."}]}, "response": "z = 1.532.", "meta": {"id": "WUA_STAT_25K_011688", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.51, μ=52.74, σ=9.39, compute z."}]}, "response": "z = -2.900.", "meta": {"id": "WUA_STAT_25K_011689", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.99, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.901.", "meta": {"id": "WUA_STAT_25K_011690", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.1, compute E and Var."}]}, "response": "E[X]=2.40; Var[X]=2.160.", "meta": {"id": "WUA_STAT_25K_011691", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.41, μ=30.78, σ=5.14, compute z."}]}, "response": "z = -3.574.", "meta": {"id": "WUA_STAT_25K_011692", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=164.37, s=36.31, n=97, compute CI."}]}, "response": "95% CI ≈ [157.14, 171.60].", "meta": {"id": "WUA_STAT_25K_011693", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.59, compute E and Var."}]}, "response": "E[X]=6.49; Var[X]=2.661.", "meta": {"id": "WUA_STAT_25K_011694", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.12, sens=0.72, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.396.", "meta": {"id": "WUA_STAT_25K_011695", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.91, s=17.19, n=143, compute CI."}]}, "response": "95% CI ≈ [175.09, 180.73].", "meta": {"id": "WUA_STAT_25K_011696", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.064, sens=0.98, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.183.", "meta": {"id": "WUA_STAT_25K_011697", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.35, s=21.35, n=72, compute CI."}]}, "response": "95% CI ≈ [28.42, 38.28].", "meta": {"id": "WUA_STAT_25K_011698", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.97, s=20.13, n=170, compute CI."}]}, "response": "95% CI ≈ [51.94, 58.00].", "meta": {"id": "WUA_STAT_25K_011699", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.38, μ=12.16, σ=8.68, compute z."}]}, "response": "z = 9.357.", "meta": {"id": "WUA_STAT_25K_011700", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.26, compute E and Var."}]}, "response": "E[X]=8.58; Var[X]=6.349.", "meta": {"id": "WUA_STAT_25K_011701", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.193, sens=0.95, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.716.", "meta": {"id": "WUA_STAT_25K_011702", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.49, μ=77.07, σ=22.0, compute z."}]}, "response": "z = 0.383.", "meta": {"id": "WUA_STAT_25K_011703", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.31, s=37.4, n=174, compute CI."}]}, "response": "95% CI ≈ [77.75, 88.87].", "meta": {"id": "WUA_STAT_25K_011704", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.117, sens=0.93, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.673.", "meta": {"id": "WUA_STAT_25K_011705", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.59, μ=74.29, σ=20.75, compute z."}]}, "response": "z = -1.961.", "meta": {"id": "WUA_STAT_25K_011706", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.47, compute E and Var."}]}, "response": "E[X]=16.45; Var[X]=8.719.", "meta": {"id": "WUA_STAT_25K_011707", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.1, s=39.02, n=194, compute CI."}]}, "response": "95% CI ≈ [63.61, 74.59].", "meta": {"id": "WUA_STAT_25K_011708", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.247, sens=0.89, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.830.", "meta": {"id": "WUA_STAT_25K_011709", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.086, sens=0.75, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.207.", "meta": {"id": "WUA_STAT_25K_011710", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.72, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.258.", "meta": {"id": "WUA_STAT_25K_011711", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.93, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.217.", "meta": {"id": "WUA_STAT_25K_011712", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.49, μ=62.75, σ=17.98, compute z."}]}, "response": "z = -1.961.", "meta": {"id": "WUA_STAT_25K_011713", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.64, compute E and Var."}]}, "response": "E[X]=13.44; Var[X]=4.838.", "meta": {"id": "WUA_STAT_25K_011714", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.69, compute E and Var."}]}, "response": "E[X]=6.90; Var[X]=2.139.", "meta": {"id": "WUA_STAT_25K_011715", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.205, sens=0.83, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.425.", "meta": {"id": "WUA_STAT_25K_011716", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.6, compute E and Var."}]}, "response": "E[X]=22.80; Var[X]=9.120.", "meta": {"id": "WUA_STAT_25K_011717", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.3, compute E and Var."}]}, "response": "E[X]=11.40; Var[X]=7.980.", "meta": {"id": "WUA_STAT_25K_011718", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.21, compute E and Var."}]}, "response": "E[X]=1.68; Var[X]=1.327.", "meta": {"id": "WUA_STAT_25K_011719", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.45, μ=60.97, σ=1.76, compute z."}]}, "response": "z = -29.273.", "meta": {"id": "WUA_STAT_25K_011720", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.25, s=33.46, n=40, compute CI."}]}, "response": "95% CI ≈ [161.88, 182.62].", "meta": {"id": "WUA_STAT_25K_011721", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.71, μ=39.53, σ=17.94, compute z."}]}, "response": "z = -1.941.", "meta": {"id": "WUA_STAT_25K_011722", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.39, s=12.27, n=136, compute CI."}]}, "response": "95% CI ≈ [35.33, 39.45].", "meta": {"id": "WUA_STAT_25K_011723", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.39, compute E and Var."}]}, "response": "E[X]=10.53; Var[X]=6.423.", "meta": {"id": "WUA_STAT_25K_011724", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.43, μ=45.69, σ=12.09, compute z."}]}, "response": "z = 0.806.", "meta": {"id": "WUA_STAT_25K_011725", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.8, compute E and Var."}]}, "response": "E[X]=4.80; Var[X]=0.960.", "meta": {"id": "WUA_STAT_25K_011726", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.24, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=6.384.", "meta": {"id": "WUA_STAT_25K_011727", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.53, compute E and Var."}]}, "response": "E[X]=12.72; Var[X]=5.978.", "meta": {"id": "WUA_STAT_25K_011728", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.153, sens=0.97, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.443.", "meta": {"id": "WUA_STAT_25K_011729", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.43, μ=83.17, σ=21.32, compute z."}]}, "response": "z = -3.553.", "meta": {"id": "WUA_STAT_25K_011730", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.207, sens=0.94, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.804.", "meta": {"id": "WUA_STAT_25K_011731", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.78, compute E and Var."}]}, "response": "E[X]=9.36; Var[X]=2.059.", "meta": {"id": "WUA_STAT_25K_011732", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.29, compute E and Var."}]}, "response": "E[X]=5.80; Var[X]=4.118.", "meta": {"id": "WUA_STAT_25K_011733", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.46, compute E and Var."}]}, "response": "E[X]=16.56; Var[X]=8.942.", "meta": {"id": "WUA_STAT_25K_011734", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.77, s=13.41, n=17, compute CI."}]}, "response": "95% CI ≈ [113.40, 126.14].", "meta": {"id": "WUA_STAT_25K_011735", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.72, s=9.98, n=46, compute CI."}]}, "response": "95% CI ≈ [76.84, 82.60].", "meta": {"id": "WUA_STAT_25K_011736", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.17, μ=91.81, σ=3.52, compute z."}]}, "response": "z = -3.307.", "meta": {"id": "WUA_STAT_25K_011737", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.78, compute E and Var."}]}, "response": "E[X]=25.74; Var[X]=5.663.", "meta": {"id": "WUA_STAT_25K_011738", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.75, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.667.", "meta": {"id": "WUA_STAT_25K_011739", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.11, μ=65.84, σ=23.88, compute z."}]}, "response": "z = 0.263.", "meta": {"id": "WUA_STAT_25K_011740", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.88, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.263.", "meta": {"id": "WUA_STAT_25K_011741", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.56, compute E and Var."}]}, "response": "E[X]=14.00; Var[X]=6.160.", "meta": {"id": "WUA_STAT_25K_011742", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.19, s=16.87, n=69, compute CI."}]}, "response": "95% CI ≈ [70.21, 78.17].", "meta": {"id": "WUA_STAT_25K_011743", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.47, s=14.31, n=22, compute CI."}]}, "response": "95% CI ≈ [176.49, 188.45].", "meta": {"id": "WUA_STAT_25K_011744", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.222, sens=0.88, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.464.", "meta": {"id": "WUA_STAT_25K_011745", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.42, μ=48.21, σ=13.32, compute z."}]}, "response": "z = -2.312.", "meta": {"id": "WUA_STAT_25K_011746", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.96, μ=12.29, σ=4.79, compute z."}]}, "response": "z = 4.942.", "meta": {"id": "WUA_STAT_25K_011747", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.78, compute E and Var."}]}, "response": "E[X]=14.04; Var[X]=3.089.", "meta": {"id": "WUA_STAT_25K_011748", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=102.12, s=7.94, n=13, compute CI."}]}, "response": "95% CI ≈ [97.80, 106.44].", "meta": {"id": "WUA_STAT_25K_011749", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.86, μ=88.02, σ=13.96, compute z."}]}, "response": "z = -3.307.", "meta": {"id": "WUA_STAT_25K_011750", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.42, compute E and Var."}]}, "response": "E[X]=4.20; Var[X]=2.436.", "meta": {"id": "WUA_STAT_25K_011751", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.84, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.599.", "meta": {"id": "WUA_STAT_25K_011752", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.131, sens=0.91, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.774.", "meta": {"id": "WUA_STAT_25K_011753", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.67, s=8.15, n=148, compute CI."}]}, "response": "95% CI ≈ [176.36, 178.98].", "meta": {"id": "WUA_STAT_25K_011754", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.89, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.326.", "meta": {"id": "WUA_STAT_25K_011755", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.33, μ=25.49, σ=6.27, compute z."}]}, "response": "z = 11.777.", "meta": {"id": "WUA_STAT_25K_011756", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.94, s=6.52, n=190, compute CI."}]}, "response": "95% CI ≈ [58.01, 59.87].", "meta": {"id": "WUA_STAT_25K_011757", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.65, s=19.38, n=73, compute CI."}]}, "response": "95% CI ≈ [86.20, 95.10].", "meta": {"id": "WUA_STAT_25K_011758", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.067, sens=0.84, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.172.", "meta": {"id": "WUA_STAT_25K_011759", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.55, s=2.41, n=114, compute CI."}]}, "response": "95% CI ≈ [15.11, 15.99].", "meta": {"id": "WUA_STAT_25K_011760", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.78, s=15.57, n=107, compute CI."}]}, "response": "95% CI ≈ [114.83, 120.73].", "meta": {"id": "WUA_STAT_25K_011761", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.1, s=24.06, n=187, compute CI."}]}, "response": "95% CI ≈ [163.65, 170.55].", "meta": {"id": "WUA_STAT_25K_011762", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.51, compute E and Var."}]}, "response": "E[X]=19.38; Var[X]=9.496.", "meta": {"id": "WUA_STAT_25K_011763", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.2, s=30.88, n=193, compute CI."}]}, "response": "95% CI ≈ [154.84, 163.56].", "meta": {"id": "WUA_STAT_25K_011764", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.38, μ=32.76, σ=15.37, compute z."}]}, "response": "z = 2.318.", "meta": {"id": "WUA_STAT_25K_011765", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.77, compute E and Var."}]}, "response": "E[X]=24.64; Var[X]=5.667.", "meta": {"id": "WUA_STAT_25K_011766", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.9, μ=40.96, σ=7.73, compute z."}]}, "response": "z = 1.545.", "meta": {"id": "WUA_STAT_25K_011767", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.131, sens=0.73, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.290.", "meta": {"id": "WUA_STAT_25K_011768", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.78, μ=82.0, σ=2.0, compute z."}]}, "response": "z = -4.110.", "meta": {"id": "WUA_STAT_25K_011769", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.05, compute E and Var."}]}, "response": "E[X]=1.85; Var[X]=1.758.", "meta": {"id": "WUA_STAT_25K_011770", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.66, compute E and Var."}]}, "response": "E[X]=10.56; Var[X]=3.590.", "meta": {"id": "WUA_STAT_25K_011771", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.246, sens=0.84, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.617.", "meta": {"id": "WUA_STAT_25K_011772", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.016, sens=0.95, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.060.", "meta": {"id": "WUA_STAT_25K_011773", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.93, μ=84.7, σ=9.98, compute z."}]}, "response": "z = -7.492.", "meta": {"id": "WUA_STAT_25K_011774", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.43, s=39.46, n=141, compute CI."}]}, "response": "95% CI ≈ [51.92, 64.94].", "meta": {"id": "WUA_STAT_25K_011775", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.15, compute E and Var."}]}, "response": "E[X]=0.90; Var[X]=0.765.", "meta": {"id": "WUA_STAT_25K_011776", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.74, compute E and Var."}]}, "response": "E[X]=14.06; Var[X]=3.656.", "meta": {"id": "WUA_STAT_25K_011777", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.81, μ=51.51, σ=15.42, compute z."}]}, "response": "z = 0.992.", "meta": {"id": "WUA_STAT_25K_011778", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.241, sens=0.88, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.608.", "meta": {"id": "WUA_STAT_25K_011779", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.037, sens=0.79, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.117.", "meta": {"id": "WUA_STAT_25K_011780", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.95, μ=97.02, σ=7.83, compute z."}]}, "response": "z = -6.650.", "meta": {"id": "WUA_STAT_25K_011781", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.3, compute E and Var."}]}, "response": "E[X]=3.90; Var[X]=2.730.", "meta": {"id": "WUA_STAT_25K_011782", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.2, μ=12.78, σ=3.61, compute z."}]}, "response": "z = 9.812.", "meta": {"id": "WUA_STAT_25K_011783", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.49, compute E and Var."}]}, "response": "E[X]=14.21; Var[X]=7.247.", "meta": {"id": "WUA_STAT_25K_011784", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.22, compute E and Var."}]}, "response": "E[X]=1.76; Var[X]=1.373.", "meta": {"id": "WUA_STAT_25K_011785", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.79, μ=37.39, σ=19.59, compute z."}]}, "response": "z = 0.327.", "meta": {"id": "WUA_STAT_25K_011786", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.44, compute E and Var."}]}, "response": "E[X]=17.16; Var[X]=9.610.", "meta": {"id": "WUA_STAT_25K_011787", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.96, μ=90.71, σ=2.53, compute z."}]}, "response": "z = 1.285.", "meta": {"id": "WUA_STAT_25K_011788", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.58, s=9.22, n=186, compute CI."}]}, "response": "95% CI ≈ [54.25, 56.91].", "meta": {"id": "WUA_STAT_25K_011789", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.46, compute E and Var."}]}, "response": "E[X]=7.36; Var[X]=3.974.", "meta": {"id": "WUA_STAT_25K_011790", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.36, compute E and Var."}]}, "response": "E[X]=8.64; Var[X]=5.530.", "meta": {"id": "WUA_STAT_25K_011791", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.78, μ=59.64, σ=10.99, compute z."}]}, "response": "z = 2.470.", "meta": {"id": "WUA_STAT_25K_011792", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.227, sens=0.98, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.567.", "meta": {"id": "WUA_STAT_25K_011793", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.68, compute E and Var."}]}, "response": "E[X]=14.28; Var[X]=4.570.", "meta": {"id": "WUA_STAT_25K_011794", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.35, s=29.29, n=30, compute CI."}]}, "response": "95% CI ≈ [119.87, 140.83].", "meta": {"id": "WUA_STAT_25K_011795", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=81.91, s=16.23, n=184, compute CI."}]}, "response": "95% CI ≈ [79.56, 84.26].", "meta": {"id": "WUA_STAT_25K_011796", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.09, compute E and Var."}]}, "response": "E[X]=2.79; Var[X]=2.539.", "meta": {"id": "WUA_STAT_25K_011797", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.88, μ=85.34, σ=14.85, compute z."}]}, "response": "z = 0.642.", "meta": {"id": "WUA_STAT_25K_011798", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.158, sens=0.88, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.355.", "meta": {"id": "WUA_STAT_25K_011799", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.5, s=38.32, n=96, compute CI."}]}, "response": "95% CI ≈ [112.83, 128.17].", "meta": {"id": "WUA_STAT_25K_011800", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.54, compute E and Var."}]}, "response": "E[X]=4.86; Var[X]=2.236.", "meta": {"id": "WUA_STAT_25K_011801", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.53, s=30.14, n=127, compute CI."}]}, "response": "95% CI ≈ [167.29, 177.77].", "meta": {"id": "WUA_STAT_25K_011802", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.73, μ=70.12, σ=4.19, compute z."}]}, "response": "z = -15.368.", "meta": {"id": "WUA_STAT_25K_011803", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.68, compute E and Var."}]}, "response": "E[X]=23.12; Var[X]=7.398.", "meta": {"id": "WUA_STAT_25K_011804", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=149.6, s=20.11, n=185, compute CI."}]}, "response": "95% CI ≈ [146.70, 152.50].", "meta": {"id": "WUA_STAT_25K_011805", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.0, s=14.89, n=93, compute CI."}]}, "response": "95% CI ≈ [82.97, 89.03].", "meta": {"id": "WUA_STAT_25K_011806", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.83, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.804.", "meta": {"id": "WUA_STAT_25K_011807", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.24, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=5.472.", "meta": {"id": "WUA_STAT_25K_011808", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.038, sens=0.96, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.275.", "meta": {"id": "WUA_STAT_25K_011809", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.06, compute E and Var."}]}, "response": "E[X]=1.98; Var[X]=1.861.", "meta": {"id": "WUA_STAT_25K_011810", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.78, s=26.73, n=137, compute CI."}]}, "response": "95% CI ≈ [148.30, 157.26].", "meta": {"id": "WUA_STAT_25K_011811", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.24, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=6.384.", "meta": {"id": "WUA_STAT_25K_011812", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.47, s=22.05, n=104, compute CI."}]}, "response": "95% CI ≈ [88.23, 96.71].", "meta": {"id": "WUA_STAT_25K_011813", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.18, compute E and Var."}]}, "response": "E[X]=1.80; Var[X]=1.476.", "meta": {"id": "WUA_STAT_25K_011814", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.36, μ=93.89, σ=13.36, compute z."}]}, "response": "z = -4.381.", "meta": {"id": "WUA_STAT_25K_011815", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.93, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.428.", "meta": {"id": "WUA_STAT_25K_011816", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=183.23, s=14.78, n=199, compute CI."}]}, "response": "95% CI ≈ [181.18, 185.28].", "meta": {"id": "WUA_STAT_25K_011817", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.29, μ=94.02, σ=4.48, compute z."}]}, "response": "z = -2.842.", "meta": {"id": "WUA_STAT_25K_011818", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.76, μ=81.7, σ=1.28, compute z."}]}, "response": "z = 1.609.", "meta": {"id": "WUA_STAT_25K_011819", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.81, s=14.32, n=85, compute CI."}]}, "response": "95% CI ≈ [176.77, 182.85].", "meta": {"id": "WUA_STAT_25K_011820", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.75, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.388.", "meta": {"id": "WUA_STAT_25K_011821", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.62, μ=60.11, σ=11.45, compute z."}]}, "response": "z = -3.536.", "meta": {"id": "WUA_STAT_25K_011822", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.48, s=15.9, n=55, compute CI."}]}, "response": "95% CI ≈ [168.28, 176.68].", "meta": {"id": "WUA_STAT_25K_011823", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.4, μ=91.25, σ=11.33, compute z."}]}, "response": "z = -3.341.", "meta": {"id": "WUA_STAT_25K_011824", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.24, μ=20.16, σ=23.19, compute z."}]}, "response": "z = -0.643.", "meta": {"id": "WUA_STAT_25K_011825", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.01, μ=41.1, σ=5.39, compute z."}]}, "response": "z = -5.583.", "meta": {"id": "WUA_STAT_25K_011826", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.62, compute E and Var."}]}, "response": "E[X]=13.02; Var[X]=4.948.", "meta": {"id": "WUA_STAT_25K_011827", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.05, s=30.38, n=122, compute CI."}]}, "response": "95% CI ≈ [144.66, 155.44].", "meta": {"id": "WUA_STAT_25K_011828", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.5, μ=71.43, σ=12.04, compute z."}]}, "response": "z = 1.667.", "meta": {"id": "WUA_STAT_25K_011829", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.08, μ=60.33, σ=16.18, compute z."}]}, "response": "z = 0.232.", "meta": {"id": "WUA_STAT_25K_011830", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.184, sens=0.78, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.575.", "meta": {"id": "WUA_STAT_25K_011831", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.12, s=5.0, n=115, compute CI."}]}, "response": "95% CI ≈ [98.21, 100.03].", "meta": {"id": "WUA_STAT_25K_011832", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.04, s=38.69, n=67, compute CI."}]}, "response": "95% CI ≈ [43.78, 62.30].", "meta": {"id": "WUA_STAT_25K_011833", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.66, compute E and Var."}]}, "response": "E[X]=11.22; Var[X]=3.815.", "meta": {"id": "WUA_STAT_25K_011834", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.55, s=7.19, n=36, compute CI."}]}, "response": "95% CI ≈ [74.20, 78.90].", "meta": {"id": "WUA_STAT_25K_011835", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.87, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.480.", "meta": {"id": "WUA_STAT_25K_011836", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.29, μ=77.02, σ=24.15, compute z."}]}, "response": "z = -2.432.", "meta": {"id": "WUA_STAT_25K_011837", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=94.37, s=31.05, n=76, compute CI."}]}, "response": "95% CI ≈ [87.39, 101.35].", "meta": {"id": "WUA_STAT_25K_011838", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.8, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.259.", "meta": {"id": "WUA_STAT_25K_011839", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.4, s=10.62, n=62, compute CI."}]}, "response": "95% CI ≈ [130.76, 136.04].", "meta": {"id": "WUA_STAT_25K_011840", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.69, compute E and Var."}]}, "response": "E[X]=26.22; Var[X]=8.128.", "meta": {"id": "WUA_STAT_25K_011841", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.16, μ=1.31, σ=10.14, compute z."}]}, "response": "z = 6.691.", "meta": {"id": "WUA_STAT_25K_011842", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.73, compute E and Var."}]}, "response": "E[X]=21.17; Var[X]=5.716.", "meta": {"id": "WUA_STAT_25K_011843", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.097, sens=0.85, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.293.", "meta": {"id": "WUA_STAT_25K_011844", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.82, s=36.09, n=84, compute CI."}]}, "response": "95% CI ≈ [122.10, 137.54].", "meta": {"id": "WUA_STAT_25K_011845", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.22, μ=47.09, σ=14.21, compute z."}]}, "response": "z = 0.291.", "meta": {"id": "WUA_STAT_25K_011846", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.229, sens=0.72, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.482.", "meta": {"id": "WUA_STAT_25K_011847", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.053, sens=0.76, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.221.", "meta": {"id": "WUA_STAT_25K_011848", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.43, compute E and Var."}]}, "response": "E[X]=7.74; Var[X]=4.412.", "meta": {"id": "WUA_STAT_25K_011849", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.89, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.114.", "meta": {"id": "WUA_STAT_25K_011850", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.55, compute E and Var."}]}, "response": "E[X]=3.85; Var[X]=1.733.", "meta": {"id": "WUA_STAT_25K_011851", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.237, sens=0.94, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.582.", "meta": {"id": "WUA_STAT_25K_011852", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.78, s=23.1, n=46, compute CI."}]}, "response": "95% CI ≈ [78.10, 91.46].", "meta": {"id": "WUA_STAT_25K_011853", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.08, μ=58.67, σ=18.55, compute z."}]}, "response": "z = -2.188.", "meta": {"id": "WUA_STAT_25K_011854", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.13, s=36.68, n=188, compute CI."}]}, "response": "95% CI ≈ [138.89, 149.37].", "meta": {"id": "WUA_STAT_25K_011855", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.192, sens=0.79, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.949.", "meta": {"id": "WUA_STAT_25K_011856", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.98, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.546.", "meta": {"id": "WUA_STAT_25K_011857", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.212, sens=0.75, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.428.", "meta": {"id": "WUA_STAT_25K_011858", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.3, compute E and Var."}]}, "response": "E[X]=4.80; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_011859", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.32, s=15.12, n=194, compute CI."}]}, "response": "95% CI ≈ [23.19, 27.45].", "meta": {"id": "WUA_STAT_25K_011860", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.166, sens=0.79, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.547.", "meta": {"id": "WUA_STAT_25K_011861", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.09, μ=57.43, σ=4.74, compute z."}]}, "response": "z = 1.616.", "meta": {"id": "WUA_STAT_25K_011862", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.47, μ=98.94, σ=20.83, compute z."}]}, "response": "z = -0.551.", "meta": {"id": "WUA_STAT_25K_011863", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.125, sens=0.71, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.480.", "meta": {"id": "WUA_STAT_25K_011864", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=141.84, s=28.65, n=79, compute CI."}]}, "response": "95% CI ≈ [135.52, 148.16].", "meta": {"id": "WUA_STAT_25K_011865", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.21, s=32.43, n=139, compute CI."}]}, "response": "95% CI ≈ [86.82, 97.60].", "meta": {"id": "WUA_STAT_25K_011866", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.76, compute E and Var."}]}, "response": "E[X]=16.72; Var[X]=4.013.", "meta": {"id": "WUA_STAT_25K_011867", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.02, μ=72.37, σ=8.03, compute z."}]}, "response": "z = -2.285.", "meta": {"id": "WUA_STAT_25K_011868", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.35, s=38.94, n=97, compute CI."}]}, "response": "95% CI ≈ [131.60, 147.10].", "meta": {"id": "WUA_STAT_25K_011869", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.56, μ=36.42, σ=21.49, compute z."}]}, "response": "z = 0.565.", "meta": {"id": "WUA_STAT_25K_011870", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.182, sens=0.98, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.456.", "meta": {"id": "WUA_STAT_25K_011871", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.36, s=1.62, n=102, compute CI."}]}, "response": "95% CI ≈ [130.05, 130.67].", "meta": {"id": "WUA_STAT_25K_011872", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.29, μ=55.52, σ=11.56, compute z."}]}, "response": "z = -0.366.", "meta": {"id": "WUA_STAT_25K_011873", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.76, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.684.", "meta": {"id": "WUA_STAT_25K_011874", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.46, compute E and Var."}]}, "response": "E[X]=17.02; Var[X]=9.191.", "meta": {"id": "WUA_STAT_25K_011875", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.6, μ=54.54, σ=12.78, compute z."}]}, "response": "z = -0.230.", "meta": {"id": "WUA_STAT_25K_011876", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.22, s=1.05, n=109, compute CI."}]}, "response": "95% CI ≈ [180.02, 180.42].", "meta": {"id": "WUA_STAT_25K_011877", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.83, μ=69.22, σ=20.91, compute z."}]}, "response": "z = 0.029.", "meta": {"id": "WUA_STAT_25K_011878", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.61, compute E and Var."}]}, "response": "E[X]=20.13; Var[X]=7.851.", "meta": {"id": "WUA_STAT_25K_011879", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.07, μ=37.25, σ=3.08, compute z."}]}, "response": "z = -1.682.", "meta": {"id": "WUA_STAT_25K_011880", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.17, sens=0.87, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.748.", "meta": {"id": "WUA_STAT_25K_011881", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.08, μ=73.36, σ=3.95, compute z."}]}, "response": "z = -3.362.", "meta": {"id": "WUA_STAT_25K_011882", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.092, sens=0.82, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.806.", "meta": {"id": "WUA_STAT_25K_011883", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.97, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.390.", "meta": {"id": "WUA_STAT_25K_011884", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.62, μ=68.7, σ=17.12, compute z."}]}, "response": "z = 0.054.", "meta": {"id": "WUA_STAT_25K_011885", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.73, compute E and Var."}]}, "response": "E[X]=6.57; Var[X]=1.774.", "meta": {"id": "WUA_STAT_25K_011886", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.39, compute E and Var."}]}, "response": "E[X]=11.31; Var[X]=6.899.", "meta": {"id": "WUA_STAT_25K_011887", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.25, s=9.33, n=80, compute CI."}]}, "response": "95% CI ≈ [121.21, 125.29].", "meta": {"id": "WUA_STAT_25K_011888", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.43, compute E and Var."}]}, "response": "E[X]=2.58; Var[X]=1.471.", "meta": {"id": "WUA_STAT_25K_011889", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.43, compute E and Var."}]}, "response": "E[X]=12.47; Var[X]=7.108.", "meta": {"id": "WUA_STAT_25K_011890", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.148, sens=0.9, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.528.", "meta": {"id": "WUA_STAT_25K_011891", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.62, compute E and Var."}]}, "response": "E[X]=7.44; Var[X]=2.827.", "meta": {"id": "WUA_STAT_25K_011892", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.11, μ=60.62, σ=2.57, compute z."}]}, "response": "z = 10.696.", "meta": {"id": "WUA_STAT_25K_011893", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.22, μ=25.54, σ=6.68, compute z."}]}, "response": "z = 7.587.", "meta": {"id": "WUA_STAT_25K_011894", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.171, sens=0.98, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.468.", "meta": {"id": "WUA_STAT_25K_011895", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.06, μ=21.44, σ=14.76, compute z."}]}, "response": "z = 1.600.", "meta": {"id": "WUA_STAT_25K_011896", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=52.58, s=30.18, n=186, compute CI."}]}, "response": "95% CI ≈ [48.24, 56.92].", "meta": {"id": "WUA_STAT_25K_011897", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.4, compute E and Var."}]}, "response": "E[X]=12.40; Var[X]=7.440.", "meta": {"id": "WUA_STAT_25K_011898", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.38, s=31.22, n=79, compute CI."}]}, "response": "95% CI ≈ [47.50, 61.26].", "meta": {"id": "WUA_STAT_25K_011899", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.55, μ=10.35, σ=16.26, compute z."}]}, "response": "z = 2.165.", "meta": {"id": "WUA_STAT_25K_011900", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.2, s=9.23, n=66, compute CI."}]}, "response": "95% CI ≈ [173.97, 178.43].", "meta": {"id": "WUA_STAT_25K_011901", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.249, sens=0.97, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.668.", "meta": {"id": "WUA_STAT_25K_011902", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.37, μ=25.67, σ=21.71, compute z."}]}, "response": "z = 3.303.", "meta": {"id": "WUA_STAT_25K_011903", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.31, μ=3.01, σ=21.45, compute z."}]}, "response": "z = 3.277.", "meta": {"id": "WUA_STAT_25K_011904", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.43, s=38.47, n=164, compute CI."}]}, "response": "95% CI ≈ [134.54, 146.32].", "meta": {"id": "WUA_STAT_25K_011905", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.72, compute E and Var."}]}, "response": "E[X]=12.24; Var[X]=3.427.", "meta": {"id": "WUA_STAT_25K_011906", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.93, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.466.", "meta": {"id": "WUA_STAT_25K_011907", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=91.75, s=28.81, n=32, compute CI."}]}, "response": "95% CI ≈ [81.77, 101.73].", "meta": {"id": "WUA_STAT_25K_011908", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.51, s=13.31, n=191, compute CI."}]}, "response": "95% CI ≈ [125.62, 129.40].", "meta": {"id": "WUA_STAT_25K_011909", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.067, sens=0.83, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.176.", "meta": {"id": "WUA_STAT_25K_011910", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.8, s=20.14, n=73, compute CI."}]}, "response": "95% CI ≈ [172.18, 181.42].", "meta": {"id": "WUA_STAT_25K_011911", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.44, s=15.93, n=150, compute CI."}]}, "response": "95% CI ≈ [27.89, 32.99].", "meta": {"id": "WUA_STAT_25K_011912", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.51, μ=71.66, σ=20.71, compute z."}]}, "response": "z = -0.876.", "meta": {"id": "WUA_STAT_25K_011913", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.19, compute E and Var."}]}, "response": "E[X]=1.33; Var[X]=1.077.", "meta": {"id": "WUA_STAT_25K_011914", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.69, μ=83.66, σ=3.91, compute z."}]}, "response": "z = -14.826.", "meta": {"id": "WUA_STAT_25K_011915", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.78, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.535.", "meta": {"id": "WUA_STAT_25K_011916", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.065, sens=0.9, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.173.", "meta": {"id": "WUA_STAT_25K_011917", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=104.86, s=6.68, n=79, compute CI."}]}, "response": "95% CI ≈ [103.39, 106.33].", "meta": {"id": "WUA_STAT_25K_011918", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.96, μ=64.24, σ=21.85, compute z."}]}, "response": "z = -1.935.", "meta": {"id": "WUA_STAT_25K_011919", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.93, s=38.35, n=135, compute CI."}]}, "response": "95% CI ≈ [138.46, 151.40].", "meta": {"id": "WUA_STAT_25K_011920", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.051, sens=0.74, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.799.", "meta": {"id": "WUA_STAT_25K_011921", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.115, sens=0.79, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.328.", "meta": {"id": "WUA_STAT_25K_011922", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.141, sens=0.99, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.489.", "meta": {"id": "WUA_STAT_25K_011923", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.13, compute E and Var."}]}, "response": "E[X]=2.47; Var[X]=2.149.", "meta": {"id": "WUA_STAT_25K_011924", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.32, compute E and Var."}]}, "response": "E[X]=3.52; Var[X]=2.394.", "meta": {"id": "WUA_STAT_25K_011925", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.49, compute E and Var."}]}, "response": "E[X]=11.27; Var[X]=5.748.", "meta": {"id": "WUA_STAT_25K_011926", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.43, s=37.49, n=65, compute CI."}]}, "response": "95% CI ≈ [16.32, 34.54].", "meta": {"id": "WUA_STAT_25K_011927", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.077, sens=0.93, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.463.", "meta": {"id": "WUA_STAT_25K_011928", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.25, sens=0.9, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.588.", "meta": {"id": "WUA_STAT_25K_011929", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.238, sens=0.79, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.565.", "meta": {"id": "WUA_STAT_25K_011930", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.72, s=15.58, n=175, compute CI."}]}, "response": "95% CI ≈ [29.41, 34.03].", "meta": {"id": "WUA_STAT_25K_011931", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.68, compute E and Var."}]}, "response": "E[X]=14.28; Var[X]=4.570.", "meta": {"id": "WUA_STAT_25K_011932", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.225, sens=0.99, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.762.", "meta": {"id": "WUA_STAT_25K_011933", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.6, compute E and Var."}]}, "response": "E[X]=6.00; Var[X]=2.400.", "meta": {"id": "WUA_STAT_25K_011934", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.76, s=12.26, n=25, compute CI."}]}, "response": "95% CI ≈ [146.95, 156.57].", "meta": {"id": "WUA_STAT_25K_011935", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.053, sens=0.73, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.151.", "meta": {"id": "WUA_STAT_25K_011936", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.093, sens=0.75, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.278.", "meta": {"id": "WUA_STAT_25K_011937", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.018, sens=0.92, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.252.", "meta": {"id": "WUA_STAT_25K_011938", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.186, sens=0.79, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.693.", "meta": {"id": "WUA_STAT_25K_011939", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=116.27, s=36.04, n=15, compute CI."}]}, "response": "95% CI ≈ [98.03, 134.51].", "meta": {"id": "WUA_STAT_25K_011940", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=143.38, s=36.18, n=175, compute CI."}]}, "response": "95% CI ≈ [138.02, 148.74].", "meta": {"id": "WUA_STAT_25K_011941", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.39, compute E and Var."}]}, "response": "E[X]=9.75; Var[X]=5.947.", "meta": {"id": "WUA_STAT_25K_011942", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.29, s=35.46, n=128, compute CI."}]}, "response": "95% CI ≈ [86.15, 98.43].", "meta": {"id": "WUA_STAT_25K_011943", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.26, compute E and Var."}]}, "response": "E[X]=3.38; Var[X]=2.501.", "meta": {"id": "WUA_STAT_25K_011944", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.1, compute E and Var."}]}, "response": "E[X]=0.70; Var[X]=0.630.", "meta": {"id": "WUA_STAT_25K_011945", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.69, compute E and Var."}]}, "response": "E[X]=16.56; Var[X]=5.134.", "meta": {"id": "WUA_STAT_25K_011946", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.72, s=9.86, n=48, compute CI."}]}, "response": "95% CI ≈ [86.93, 92.51].", "meta": {"id": "WUA_STAT_25K_011947", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.99, s=5.01, n=143, compute CI."}]}, "response": "95% CI ≈ [119.17, 120.81].", "meta": {"id": "WUA_STAT_25K_011948", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.85, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.446.", "meta": {"id": "WUA_STAT_25K_011949", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.13, μ=13.39, σ=14.46, compute z."}]}, "response": "z = 3.509.", "meta": {"id": "WUA_STAT_25K_011950", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.141, sens=0.95, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.886.", "meta": {"id": "WUA_STAT_25K_011951", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=171.69, s=9.16, n=163, compute CI."}]}, "response": "95% CI ≈ [170.28, 173.10].", "meta": {"id": "WUA_STAT_25K_011952", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.45, compute E and Var."}]}, "response": "E[X]=14.40; Var[X]=7.920.", "meta": {"id": "WUA_STAT_25K_011953", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.15, s=5.49, n=114, compute CI."}]}, "response": "95% CI ≈ [98.14, 100.16].", "meta": {"id": "WUA_STAT_25K_011954", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.32, compute E and Var."}]}, "response": "E[X]=1.60; Var[X]=1.088.", "meta": {"id": "WUA_STAT_25K_011955", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.77, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.172.", "meta": {"id": "WUA_STAT_25K_011956", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.45, compute E and Var."}]}, "response": "E[X]=11.25; Var[X]=6.188.", "meta": {"id": "WUA_STAT_25K_011957", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=102.89, s=24.99, n=110, compute CI."}]}, "response": "95% CI ≈ [98.22, 107.56].", "meta": {"id": "WUA_STAT_25K_011958", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.66, μ=59.26, σ=15.57, compute z."}]}, "response": "z = -1.837.", "meta": {"id": "WUA_STAT_25K_011959", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.18, μ=3.59, σ=1.35, compute z."}]}, "response": "z = 19.696.", "meta": {"id": "WUA_STAT_25K_011960", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.07, compute E and Var."}]}, "response": "E[X]=2.03; Var[X]=1.888.", "meta": {"id": "WUA_STAT_25K_011961", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.45, s=28.43, n=72, compute CI."}]}, "response": "95% CI ≈ [175.88, 189.02].", "meta": {"id": "WUA_STAT_25K_011962", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.15, compute E and Var."}]}, "response": "E[X]=3.30; Var[X]=2.805.", "meta": {"id": "WUA_STAT_25K_011963", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.85, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.401.", "meta": {"id": "WUA_STAT_25K_011964", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=66.71, s=7.38, n=153, compute CI."}]}, "response": "95% CI ≈ [65.54, 67.88].", "meta": {"id": "WUA_STAT_25K_011965", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.24, μ=46.27, σ=14.36, compute z."}]}, "response": "z = 0.764.", "meta": {"id": "WUA_STAT_25K_011966", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.05, μ=49.42, σ=21.2, compute z."}]}, "response": "z = 1.209.", "meta": {"id": "WUA_STAT_25K_011967", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.02, s=2.97, n=53, compute CI."}]}, "response": "95% CI ≈ [54.22, 55.82].", "meta": {"id": "WUA_STAT_25K_011968", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.7, compute E and Var."}]}, "response": "E[X]=18.90; Var[X]=5.670.", "meta": {"id": "WUA_STAT_25K_011969", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.58, compute E and Var."}]}, "response": "E[X]=9.28; Var[X]=3.898.", "meta": {"id": "WUA_STAT_25K_011970", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.3, μ=2.84, σ=16.08, compute z."}]}, "response": "z = 2.454.", "meta": {"id": "WUA_STAT_25K_011971", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.159, sens=0.95, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.620.", "meta": {"id": "WUA_STAT_25K_011972", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.69, s=27.08, n=186, compute CI."}]}, "response": "95% CI ≈ [150.80, 158.58].", "meta": {"id": "WUA_STAT_25K_011973", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.68, μ=55.68, σ=11.14, compute z."}]}, "response": "z = -0.269.", "meta": {"id": "WUA_STAT_25K_011974", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.72, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.742.", "meta": {"id": "WUA_STAT_25K_011975", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.3, s=16.19, n=85, compute CI."}]}, "response": "95% CI ≈ [142.86, 149.74].", "meta": {"id": "WUA_STAT_25K_011976", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.51, μ=45.11, σ=21.4, compute z."}]}, "response": "z = 1.467.", "meta": {"id": "WUA_STAT_25K_011977", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.31, compute E and Var."}]}, "response": "E[X]=4.65; Var[X]=3.208.", "meta": {"id": "WUA_STAT_25K_011978", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.41, compute E and Var."}]}, "response": "E[X]=11.48; Var[X]=6.773.", "meta": {"id": "WUA_STAT_25K_011979", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.9, s=33.6, n=157, compute CI."}]}, "response": "95% CI ≈ [36.64, 47.16].", "meta": {"id": "WUA_STAT_25K_011980", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.97, μ=24.06, σ=13.19, compute z."}]}, "response": "z = 1.358.", "meta": {"id": "WUA_STAT_25K_011981", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.65, μ=51.21, σ=14.82, compute z."}]}, "response": "z = -3.412.", "meta": {"id": "WUA_STAT_25K_011982", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.78, μ=94.94, σ=21.03, compute z."}]}, "response": "z = -2.623.", "meta": {"id": "WUA_STAT_25K_011983", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.78, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.664.", "meta": {"id": "WUA_STAT_25K_011984", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.22, s=6.7, n=22, compute CI."}]}, "response": "95% CI ≈ [94.42, 100.02].", "meta": {"id": "WUA_STAT_25K_011985", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.78, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.830.", "meta": {"id": "WUA_STAT_25K_011986", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.74, μ=36.14, σ=21.01, compute z."}]}, "response": "z = 0.838.", "meta": {"id": "WUA_STAT_25K_011987", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=20.07, s=17.43, n=130, compute CI."}]}, "response": "95% CI ≈ [17.07, 23.07].", "meta": {"id": "WUA_STAT_25K_011988", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.37, μ=81.37, σ=19.23, compute z."}]}, "response": "z = -4.212.", "meta": {"id": "WUA_STAT_25K_011989", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.64, compute E and Var."}]}, "response": "E[X]=3.20; Var[X]=1.152.", "meta": {"id": "WUA_STAT_25K_011990", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.84, μ=76.39, σ=23.21, compute z."}]}, "response": "z = 0.795.", "meta": {"id": "WUA_STAT_25K_011991", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.57, compute E and Var."}]}, "response": "E[X]=4.56; Var[X]=1.961.", "meta": {"id": "WUA_STAT_25K_011992", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.6, compute E and Var."}]}, "response": "E[X]=3.60; Var[X]=1.440.", "meta": {"id": "WUA_STAT_25K_011993", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.27, μ=8.47, σ=22.06, compute z."}]}, "response": "z = 1.079.", "meta": {"id": "WUA_STAT_25K_011994", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.73, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.450.", "meta": {"id": "WUA_STAT_25K_011995", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.61, compute E and Var."}]}, "response": "E[X]=8.54; Var[X]=3.331.", "meta": {"id": "WUA_STAT_25K_011996", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.119, sens=0.85, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.284.", "meta": {"id": "WUA_STAT_25K_011997", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.91, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.765.", "meta": {"id": "WUA_STAT_25K_011998", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.21, compute E and Var."}]}, "response": "E[X]=3.36; Var[X]=2.654.", "meta": {"id": "WUA_STAT_25K_011999", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.88, s=8.59, n=120, compute CI."}]}, "response": "95% CI ≈ [56.34, 59.42].", "meta": {"id": "WUA_STAT_25K_012000", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.2, μ=28.42, σ=23.13, compute z."}]}, "response": "z = -0.312.", "meta": {"id": "WUA_STAT_25K_012001", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.21, compute E and Var."}]}, "response": "E[X]=4.83; Var[X]=3.816.", "meta": {"id": "WUA_STAT_25K_012002", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.212, sens=0.74, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.570.", "meta": {"id": "WUA_STAT_25K_012003", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=69.13, s=6.6, n=95, compute CI."}]}, "response": "95% CI ≈ [67.80, 70.46].", "meta": {"id": "WUA_STAT_25K_012004", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.04, s=1.87, n=133, compute CI."}]}, "response": "95% CI ≈ [23.72, 24.36].", "meta": {"id": "WUA_STAT_25K_012005", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.04, μ=13.36, σ=7.59, compute z."}]}, "response": "z = 7.072.", "meta": {"id": "WUA_STAT_25K_012006", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.44, compute E and Var."}]}, "response": "E[X]=7.48; Var[X]=4.189.", "meta": {"id": "WUA_STAT_25K_012007", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.39, μ=74.38, σ=16.02, compute z."}]}, "response": "z = -1.560.", "meta": {"id": "WUA_STAT_25K_012008", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.06, s=16.95, n=77, compute CI."}]}, "response": "95% CI ≈ [73.27, 80.85].", "meta": {"id": "WUA_STAT_25K_012009", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.16, μ=97.34, σ=1.41, compute z."}]}, "response": "z = -0.837.", "meta": {"id": "WUA_STAT_25K_012010", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.72, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.449.", "meta": {"id": "WUA_STAT_25K_012011", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.76, s=2.86, n=194, compute CI."}]}, "response": "95% CI ≈ [160.36, 161.16].", "meta": {"id": "WUA_STAT_25K_012012", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.92, s=23.82, n=63, compute CI."}]}, "response": "95% CI ≈ [69.04, 80.80].", "meta": {"id": "WUA_STAT_25K_012013", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.93, μ=50.0, σ=4.07, compute z."}]}, "response": "z = 1.703.", "meta": {"id": "WUA_STAT_25K_012014", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.4, compute E and Var."}]}, "response": "E[X]=4.40; Var[X]=2.640.", "meta": {"id": "WUA_STAT_25K_012015", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.52, compute E and Var."}]}, "response": "E[X]=10.92; Var[X]=5.242.", "meta": {"id": "WUA_STAT_25K_012016", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.26, compute E and Var."}]}, "response": "E[X]=2.86; Var[X]=2.116.", "meta": {"id": "WUA_STAT_25K_012017", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.87, μ=94.98, σ=11.72, compute z."}]}, "response": "z = -2.228.", "meta": {"id": "WUA_STAT_25K_012018", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.165, sens=0.82, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.384.", "meta": {"id": "WUA_STAT_25K_012019", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.09, compute E and Var."}]}, "response": "E[X]=2.25; Var[X]=2.047.", "meta": {"id": "WUA_STAT_25K_012020", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.21, s=1.27, n=107, compute CI."}]}, "response": "95% CI ≈ [113.97, 114.45].", "meta": {"id": "WUA_STAT_25K_012021", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.04, s=39.26, n=185, compute CI."}]}, "response": "95% CI ≈ [164.38, 175.70].", "meta": {"id": "WUA_STAT_25K_012022", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.26, μ=43.45, σ=21.34, compute z."}]}, "response": "z = -1.462.", "meta": {"id": "WUA_STAT_25K_012023", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.35, s=22.42, n=182, compute CI."}]}, "response": "95% CI ≈ [21.09, 27.61].", "meta": {"id": "WUA_STAT_25K_012024", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.03, μ=79.45, σ=3.66, compute z."}]}, "response": "z = -10.224.", "meta": {"id": "WUA_STAT_25K_012025", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.21, compute E and Var."}]}, "response": "E[X]=6.72; Var[X]=5.309.", "meta": {"id": "WUA_STAT_25K_012026", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.67, compute E and Var."}]}, "response": "E[X]=16.75; Var[X]=5.527.", "meta": {"id": "WUA_STAT_25K_012027", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.011, sens=0.83, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.093.", "meta": {"id": "WUA_STAT_25K_012028", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.79, μ=92.38, σ=18.48, compute z."}]}, "response": "z = -0.790.", "meta": {"id": "WUA_STAT_25K_012029", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.203, sens=0.97, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.487.", "meta": {"id": "WUA_STAT_25K_012030", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.5, μ=11.38, σ=6.0, compute z."}]}, "response": "z = -1.813.", "meta": {"id": "WUA_STAT_25K_012031", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.49, compute E and Var."}]}, "response": "E[X]=11.76; Var[X]=5.998.", "meta": {"id": "WUA_STAT_25K_012032", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.86, μ=17.5, σ=9.91, compute z."}]}, "response": "z = 1.146.", "meta": {"id": "WUA_STAT_25K_012033", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.104, sens=0.92, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.348.", "meta": {"id": "WUA_STAT_25K_012034", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=148.23, s=29.82, n=40, compute CI."}]}, "response": "95% CI ≈ [138.99, 157.47].", "meta": {"id": "WUA_STAT_25K_012035", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.78, μ=71.71, σ=7.38, compute z."}]}, "response": "z = -4.869.", "meta": {"id": "WUA_STAT_25K_012036", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=197.0, s=9.2, n=119, compute CI."}]}, "response": "95% CI ≈ [195.35, 198.65].", "meta": {"id": "WUA_STAT_25K_012037", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.91, s=30.4, n=194, compute CI."}]}, "response": "95% CI ≈ [63.63, 72.19].", "meta": {"id": "WUA_STAT_25K_012038", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.92, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.768.", "meta": {"id": "WUA_STAT_25K_012039", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.06, compute E and Var."}]}, "response": "E[X]=2.34; Var[X]=2.200.", "meta": {"id": "WUA_STAT_25K_012040", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.78, μ=12.75, σ=6.95, compute z."}]}, "response": "z = 10.220.", "meta": {"id": "WUA_STAT_25K_012041", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.087, sens=0.74, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.638.", "meta": {"id": "WUA_STAT_25K_012042", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.166, sens=0.75, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.365.", "meta": {"id": "WUA_STAT_25K_012043", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.75, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.463.", "meta": {"id": "WUA_STAT_25K_012044", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.88, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.481.", "meta": {"id": "WUA_STAT_25K_012045", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.5, compute E and Var."}]}, "response": "E[X]=6.50; Var[X]=3.250.", "meta": {"id": "WUA_STAT_25K_012046", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.09, μ=25.91, σ=12.42, compute z."}]}, "response": "z = -1.676.", "meta": {"id": "WUA_STAT_25K_012047", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.91, μ=18.02, σ=22.68, compute z."}]}, "response": "z = 0.260.", "meta": {"id": "WUA_STAT_25K_012048", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.64, μ=1.95, σ=4.44, compute z."}]}, "response": "z = 11.642.", "meta": {"id": "WUA_STAT_25K_012049", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.88, μ=90.72, σ=18.78, compute z."}]}, "response": "z = -1.695.", "meta": {"id": "WUA_STAT_25K_012050", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.63, compute E and Var."}]}, "response": "E[X]=18.90; Var[X]=6.993.", "meta": {"id": "WUA_STAT_25K_012051", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.61, compute E and Var."}]}, "response": "E[X]=20.74; Var[X]=8.089.", "meta": {"id": "WUA_STAT_25K_012052", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.5, s=28.16, n=196, compute CI."}]}, "response": "95% CI ≈ [72.56, 80.44].", "meta": {"id": "WUA_STAT_25K_012053", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.94, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.852.", "meta": {"id": "WUA_STAT_25K_012054", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.46, compute E and Var."}]}, "response": "E[X]=4.14; Var[X]=2.236.", "meta": {"id": "WUA_STAT_25K_012055", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.64, μ=12.92, σ=20.62, compute z."}]}, "response": "z = 1.635.", "meta": {"id": "WUA_STAT_25K_012056", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.64, compute E and Var."}]}, "response": "E[X]=8.32; Var[X]=2.995.", "meta": {"id": "WUA_STAT_25K_012057", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.47, compute E and Var."}]}, "response": "E[X]=10.34; Var[X]=5.480.", "meta": {"id": "WUA_STAT_25K_012058", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.69, s=35.56, n=74, compute CI."}]}, "response": "95% CI ≈ [55.59, 71.79].", "meta": {"id": "WUA_STAT_25K_012059", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.36, compute E and Var."}]}, "response": "E[X]=4.32; Var[X]=2.765.", "meta": {"id": "WUA_STAT_25K_012060", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.41, compute E and Var."}]}, "response": "E[X]=11.89; Var[X]=7.015.", "meta": {"id": "WUA_STAT_25K_012061", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.76, s=11.45, n=104, compute CI."}]}, "response": "95% CI ≈ [192.56, 196.96].", "meta": {"id": "WUA_STAT_25K_012062", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.36, μ=35.24, σ=4.44, compute z."}]}, "response": "z = 7.685.", "meta": {"id": "WUA_STAT_25K_012063", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.22, μ=32.52, σ=14.71, compute z."}]}, "response": "z = -0.632.", "meta": {"id": "WUA_STAT_25K_012064", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.25, μ=25.04, σ=14.56, compute z."}]}, "response": "z = 1.800.", "meta": {"id": "WUA_STAT_25K_012065", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.06, s=1.26, n=51, compute CI."}]}, "response": "95% CI ≈ [30.71, 31.41].", "meta": {"id": "WUA_STAT_25K_012066", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.73, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.441.", "meta": {"id": "WUA_STAT_25K_012067", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=126.64, s=36.0, n=81, compute CI."}]}, "response": "95% CI ≈ [118.80, 134.48].", "meta": {"id": "WUA_STAT_25K_012068", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.49, μ=9.49, σ=18.66, compute z."}]}, "response": "z = 1.447.", "meta": {"id": "WUA_STAT_25K_012069", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.69, s=8.8, n=161, compute CI."}]}, "response": "95% CI ≈ [62.33, 65.05].", "meta": {"id": "WUA_STAT_25K_012070", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.096, sens=0.71, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.557.", "meta": {"id": "WUA_STAT_25K_012071", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.77, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.337.", "meta": {"id": "WUA_STAT_25K_012072", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.64, s=1.37, n=23, compute CI."}]}, "response": "95% CI ≈ [62.08, 63.20].", "meta": {"id": "WUA_STAT_25K_012073", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.23, s=30.63, n=89, compute CI."}]}, "response": "95% CI ≈ [156.87, 169.59].", "meta": {"id": "WUA_STAT_25K_012074", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.29, compute E and Var."}]}, "response": "E[X]=7.83; Var[X]=5.559.", "meta": {"id": "WUA_STAT_25K_012075", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.95, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.409.", "meta": {"id": "WUA_STAT_25K_012076", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.14, compute E and Var."}]}, "response": "E[X]=0.98; Var[X]=0.843.", "meta": {"id": "WUA_STAT_25K_012077", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.6, μ=14.84, σ=7.95, compute z."}]}, "response": "z = 10.158.", "meta": {"id": "WUA_STAT_25K_012078", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.54, compute E and Var."}]}, "response": "E[X]=3.24; Var[X]=1.490.", "meta": {"id": "WUA_STAT_25K_012079", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.72, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.791.", "meta": {"id": "WUA_STAT_25K_012080", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.94, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.832.", "meta": {"id": "WUA_STAT_25K_012081", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.82, s=12.77, n=111, compute CI."}]}, "response": "95% CI ≈ [116.44, 121.20].", "meta": {"id": "WUA_STAT_25K_012082", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.54, s=14.48, n=119, compute CI."}]}, "response": "95% CI ≈ [58.94, 64.14].", "meta": {"id": "WUA_STAT_25K_012083", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=42.81, s=17.68, n=187, compute CI."}]}, "response": "95% CI ≈ [40.28, 45.34].", "meta": {"id": "WUA_STAT_25K_012084", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.45, compute E and Var."}]}, "response": "E[X]=13.05; Var[X]=7.178.", "meta": {"id": "WUA_STAT_25K_012085", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.46, s=16.59, n=148, compute CI."}]}, "response": "95% CI ≈ [43.79, 49.13].", "meta": {"id": "WUA_STAT_25K_012086", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.11, s=19.07, n=175, compute CI."}]}, "response": "95% CI ≈ [167.28, 172.94].", "meta": {"id": "WUA_STAT_25K_012087", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.47, s=5.88, n=128, compute CI."}]}, "response": "95% CI ≈ [151.45, 153.49].", "meta": {"id": "WUA_STAT_25K_012088", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.06, s=33.6, n=120, compute CI."}]}, "response": "95% CI ≈ [105.05, 117.07].", "meta": {"id": "WUA_STAT_25K_012089", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.26, s=36.49, n=66, compute CI."}]}, "response": "95% CI ≈ [100.46, 118.06].", "meta": {"id": "WUA_STAT_25K_012090", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.156, sens=0.9, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.675.", "meta": {"id": "WUA_STAT_25K_012091", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.054, sens=0.72, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.186.", "meta": {"id": "WUA_STAT_25K_012092", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.01, μ=67.06, σ=14.76, compute z."}]}, "response": "z = -3.594.", "meta": {"id": "WUA_STAT_25K_012093", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.062, sens=0.74, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.189.", "meta": {"id": "WUA_STAT_25K_012094", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.042, sens=0.83, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.168.", "meta": {"id": "WUA_STAT_25K_012095", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.15, s=30.07, n=97, compute CI."}]}, "response": "95% CI ≈ [35.17, 47.13].", "meta": {"id": "WUA_STAT_25K_012096", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.37, μ=8.74, σ=7.37, compute z."}]}, "response": "z = 2.664.", "meta": {"id": "WUA_STAT_25K_012097", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.39, μ=79.73, σ=18.79, compute z."}]}, "response": "z = -3.850.", "meta": {"id": "WUA_STAT_25K_012098", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.217, sens=0.76, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.526.", "meta": {"id": "WUA_STAT_25K_012099", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=143.79, s=39.92, n=57, compute CI."}]}, "response": "95% CI ≈ [133.43, 154.15].", "meta": {"id": "WUA_STAT_25K_012100", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.22, s=30.67, n=142, compute CI."}]}, "response": "95% CI ≈ [6.18, 16.26].", "meta": {"id": "WUA_STAT_25K_012101", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.76, compute E and Var."}]}, "response": "E[X]=30.40; Var[X]=7.296.", "meta": {"id": "WUA_STAT_25K_012102", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.32, compute E and Var."}]}, "response": "E[X]=12.48; Var[X]=8.486.", "meta": {"id": "WUA_STAT_25K_012103", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.43, μ=94.33, σ=1.02, compute z."}]}, "response": "z = -0.882.", "meta": {"id": "WUA_STAT_25K_012104", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.64, μ=64.69, σ=9.15, compute z."}]}, "response": "z = -1.098.", "meta": {"id": "WUA_STAT_25K_012105", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=94.23, s=27.34, n=57, compute CI."}]}, "response": "95% CI ≈ [87.13, 101.33].", "meta": {"id": "WUA_STAT_25K_012106", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=192.05, s=6.57, n=30, compute CI."}]}, "response": "95% CI ≈ [189.70, 194.40].", "meta": {"id": "WUA_STAT_25K_012107", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.1, μ=28.67, σ=1.14, compute z."}]}, "response": "z = -16.289.", "meta": {"id": "WUA_STAT_25K_012108", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.29, s=12.84, n=118, compute CI."}]}, "response": "95% CI ≈ [169.97, 174.61].", "meta": {"id": "WUA_STAT_25K_012109", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.77, s=9.66, n=102, compute CI."}]}, "response": "95% CI ≈ [164.90, 168.64].", "meta": {"id": "WUA_STAT_25K_012110", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.21, μ=77.46, σ=24.11, compute z."}]}, "response": "z = -0.259.", "meta": {"id": "WUA_STAT_25K_012111", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.59, μ=14.99, σ=19.54, compute z."}]}, "response": "z = 2.692.", "meta": {"id": "WUA_STAT_25K_012112", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.38, compute E and Var."}]}, "response": "E[X]=3.04; Var[X]=1.885.", "meta": {"id": "WUA_STAT_25K_012113", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.193, sens=0.88, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.501.", "meta": {"id": "WUA_STAT_25K_012114", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.35, compute E and Var."}]}, "response": "E[X]=4.20; Var[X]=2.730.", "meta": {"id": "WUA_STAT_25K_012115", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.8, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.706.", "meta": {"id": "WUA_STAT_25K_012116", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=95.04, s=2.59, n=61, compute CI."}]}, "response": "95% CI ≈ [94.39, 95.69].", "meta": {"id": "WUA_STAT_25K_012117", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.57, compute E and Var."}]}, "response": "E[X]=18.24; Var[X]=7.843.", "meta": {"id": "WUA_STAT_25K_012118", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=71.63, s=5.56, n=130, compute CI."}]}, "response": "95% CI ≈ [70.67, 72.59].", "meta": {"id": "WUA_STAT_25K_012119", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.237, sens=0.99, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.737.", "meta": {"id": "WUA_STAT_25K_012120", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.89, μ=41.1, σ=24.32, compute z."}]}, "response": "z = -0.461.", "meta": {"id": "WUA_STAT_25K_012121", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.25, compute E and Var."}]}, "response": "E[X]=7.00; Var[X]=5.250.", "meta": {"id": "WUA_STAT_25K_012122", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.46, μ=16.96, σ=6.22, compute z."}]}, "response": "z = 5.064.", "meta": {"id": "WUA_STAT_25K_012123", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.69, compute E and Var."}]}, "response": "E[X]=7.59; Var[X]=2.353.", "meta": {"id": "WUA_STAT_25K_012124", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.98, s=14.97, n=73, compute CI."}]}, "response": "95% CI ≈ [20.55, 27.41].", "meta": {"id": "WUA_STAT_25K_012125", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.32, compute E and Var."}]}, "response": "E[X]=3.20; Var[X]=2.176.", "meta": {"id": "WUA_STAT_25K_012126", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.82, μ=44.59, σ=24.41, compute z."}]}, "response": "z = 1.238.", "meta": {"id": "WUA_STAT_25K_012127", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.55, compute E and Var."}]}, "response": "E[X]=21.45; Var[X]=9.652.", "meta": {"id": "WUA_STAT_25K_012128", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.8, μ=13.97, σ=12.65, compute z."}]}, "response": "z = 5.678.", "meta": {"id": "WUA_STAT_25K_012129", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.43, compute E and Var."}]}, "response": "E[X]=9.03; Var[X]=5.147.", "meta": {"id": "WUA_STAT_25K_012130", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.15, μ=48.49, σ=14.66, compute z."}]}, "response": "z = -2.956.", "meta": {"id": "WUA_STAT_25K_012131", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.03, s=29.62, n=193, compute CI."}]}, "response": "95% CI ≈ [156.85, 165.21].", "meta": {"id": "WUA_STAT_25K_012132", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.27, s=25.63, n=28, compute CI."}]}, "response": "95% CI ≈ [156.78, 175.76].", "meta": {"id": "WUA_STAT_25K_012133", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.85, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.798.", "meta": {"id": "WUA_STAT_25K_012134", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.19, compute E and Var."}]}, "response": "E[X]=1.14; Var[X]=0.923.", "meta": {"id": "WUA_STAT_25K_012135", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.36, μ=25.58, σ=13.11, compute z."}]}, "response": "z = -0.093.", "meta": {"id": "WUA_STAT_25K_012136", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.33, μ=34.4, σ=24.59, compute z."}]}, "response": "z = 0.729.", "meta": {"id": "WUA_STAT_25K_012137", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.44, s=25.31, n=22, compute CI."}]}, "response": "95% CI ≈ [171.86, 193.02].", "meta": {"id": "WUA_STAT_25K_012138", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.42, μ=45.51, σ=22.44, compute z."}]}, "response": "z = -1.653.", "meta": {"id": "WUA_STAT_25K_012139", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.96, μ=81.87, σ=8.11, compute z."}]}, "response": "z = -2.825.", "meta": {"id": "WUA_STAT_25K_012140", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.69, compute E and Var."}]}, "response": "E[X]=3.45; Var[X]=1.070.", "meta": {"id": "WUA_STAT_25K_012141", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.23, sens=0.92, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.487.", "meta": {"id": "WUA_STAT_25K_012142", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.95, μ=18.63, σ=18.93, compute z."}]}, "response": "z = 0.387.", "meta": {"id": "WUA_STAT_25K_012143", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.99, μ=61.35, σ=3.77, compute z."}]}, "response": "z = -12.828.", "meta": {"id": "WUA_STAT_25K_012144", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.31, compute E and Var."}]}, "response": "E[X]=11.16; Var[X]=7.700.", "meta": {"id": "WUA_STAT_25K_012145", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.04, s=38.41, n=75, compute CI."}]}, "response": "95% CI ≈ [31.35, 48.73].", "meta": {"id": "WUA_STAT_25K_012146", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.48, compute E and Var."}]}, "response": "E[X]=6.72; Var[X]=3.494.", "meta": {"id": "WUA_STAT_25K_012147", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.76, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.216.", "meta": {"id": "WUA_STAT_25K_012148", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.11, s=10.63, n=80, compute CI."}]}, "response": "95% CI ≈ [172.78, 177.44].", "meta": {"id": "WUA_STAT_25K_012149", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=124.29, s=21.82, n=109, compute CI."}]}, "response": "95% CI ≈ [120.19, 128.39].", "meta": {"id": "WUA_STAT_25K_012150", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.72, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.903.", "meta": {"id": "WUA_STAT_25K_012151", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.033, sens=0.76, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.139.", "meta": {"id": "WUA_STAT_25K_012152", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.72, μ=25.59, σ=22.28, compute z."}]}, "response": "z = -0.039.", "meta": {"id": "WUA_STAT_25K_012153", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.73, μ=46.86, σ=14.62, compute z."}]}, "response": "z = 0.744.", "meta": {"id": "WUA_STAT_25K_012154", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.08, μ=14.9, σ=2.92, compute z."}]}, "response": "z = 17.870.", "meta": {"id": "WUA_STAT_25K_012155", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.57, compute E and Var."}]}, "response": "E[X]=3.99; Var[X]=1.716.", "meta": {"id": "WUA_STAT_25K_012156", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.241, sens=0.77, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.485.", "meta": {"id": "WUA_STAT_25K_012157", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.44, μ=86.8, σ=8.46, compute z."}]}, "response": "z = -5.598.", "meta": {"id": "WUA_STAT_25K_012158", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.6, μ=37.67, σ=5.75, compute z."}]}, "response": "z = 3.988.", "meta": {"id": "WUA_STAT_25K_012159", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.16, s=2.91, n=78, compute CI."}]}, "response": "95% CI ≈ [98.51, 99.81].", "meta": {"id": "WUA_STAT_25K_012160", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.024, sens=0.89, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.070.", "meta": {"id": "WUA_STAT_25K_012161", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.2, μ=4.35, σ=13.8, compute z."}]}, "response": "z = 1.946.", "meta": {"id": "WUA_STAT_25K_012162", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.159, sens=0.8, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.602.", "meta": {"id": "WUA_STAT_25K_012163", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.66, compute E and Var."}]}, "response": "E[X]=24.42; Var[X]=8.303.", "meta": {"id": "WUA_STAT_25K_012164", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.07, compute E and Var."}]}, "response": "E[X]=1.61; Var[X]=1.497.", "meta": {"id": "WUA_STAT_25K_012165", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.73, s=35.48, n=28, compute CI."}]}, "response": "95% CI ≈ [110.59, 136.87].", "meta": {"id": "WUA_STAT_25K_012166", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.52, compute E and Var."}]}, "response": "E[X]=9.88; Var[X]=4.742.", "meta": {"id": "WUA_STAT_25K_012167", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.72, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.727.", "meta": {"id": "WUA_STAT_25K_012168", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.78, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.378.", "meta": {"id": "WUA_STAT_25K_012169", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.72, μ=12.85, σ=16.9, compute z."}]}, "response": "z = 0.998.", "meta": {"id": "WUA_STAT_25K_012170", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.03, μ=72.62, σ=6.52, compute z."}]}, "response": "z = 3.590.", "meta": {"id": "WUA_STAT_25K_012171", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.82, s=32.93, n=76, compute CI."}]}, "response": "95% CI ≈ [163.42, 178.22].", "meta": {"id": "WUA_STAT_25K_012172", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.61, s=34.06, n=59, compute CI."}]}, "response": "95% CI ≈ [180.92, 198.30].", "meta": {"id": "WUA_STAT_25K_012173", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=52.12, s=30.77, n=60, compute CI."}]}, "response": "95% CI ≈ [44.33, 59.91].", "meta": {"id": "WUA_STAT_25K_012174", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.06, μ=83.96, σ=13.3, compute z."}]}, "response": "z = -3.150.", "meta": {"id": "WUA_STAT_25K_012175", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.052, sens=0.98, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.240.", "meta": {"id": "WUA_STAT_25K_012176", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.76, s=18.85, n=174, compute CI."}]}, "response": "95% CI ≈ [183.96, 189.56].", "meta": {"id": "WUA_STAT_25K_012177", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.76, μ=25.35, σ=8.22, compute z."}]}, "response": "z = 2.483.", "meta": {"id": "WUA_STAT_25K_012178", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.67, compute E and Var."}]}, "response": "E[X]=4.02; Var[X]=1.327.", "meta": {"id": "WUA_STAT_25K_012179", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.149, sens=0.73, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.402.", "meta": {"id": "WUA_STAT_25K_012180", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.45, s=16.44, n=16, compute CI."}]}, "response": "95% CI ≈ [66.39, 82.51].", "meta": {"id": "WUA_STAT_25K_012181", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.95, s=27.08, n=143, compute CI."}]}, "response": "95% CI ≈ [172.51, 181.39].", "meta": {"id": "WUA_STAT_25K_012182", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.65, s=8.93, n=92, compute CI."}]}, "response": "95% CI ≈ [111.83, 115.47].", "meta": {"id": "WUA_STAT_25K_012183", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.99, s=34.49, n=54, compute CI."}]}, "response": "95% CI ≈ [126.79, 145.19].", "meta": {"id": "WUA_STAT_25K_012184", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.65, μ=43.31, σ=16.85, compute z."}]}, "response": "z = 1.801.", "meta": {"id": "WUA_STAT_25K_012185", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.98, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.204.", "meta": {"id": "WUA_STAT_25K_012186", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.93, μ=17.77, σ=21.42, compute z."}]}, "response": "z = 1.688.", "meta": {"id": "WUA_STAT_25K_012187", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.82, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.510.", "meta": {"id": "WUA_STAT_25K_012188", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.38, s=29.24, n=70, compute CI."}]}, "response": "95% CI ≈ [180.53, 194.23].", "meta": {"id": "WUA_STAT_25K_012189", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.53, s=3.94, n=87, compute CI."}]}, "response": "95% CI ≈ [97.70, 99.36].", "meta": {"id": "WUA_STAT_25K_012190", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.8, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.323.", "meta": {"id": "WUA_STAT_25K_012191", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=100.77, s=33.31, n=107, compute CI."}]}, "response": "95% CI ≈ [94.46, 107.08].", "meta": {"id": "WUA_STAT_25K_012192", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.25, μ=13.02, σ=1.05, compute z."}]}, "response": "z = 15.457.", "meta": {"id": "WUA_STAT_25K_012193", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=68.23, s=24.66, n=156, compute CI."}]}, "response": "95% CI ≈ [64.36, 72.10].", "meta": {"id": "WUA_STAT_25K_012194", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.76, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.084.", "meta": {"id": "WUA_STAT_25K_012195", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.38, compute E and Var."}]}, "response": "E[X]=5.32; Var[X]=3.298.", "meta": {"id": "WUA_STAT_25K_012196", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.02, μ=72.54, σ=4.57, compute z."}]}, "response": "z = -3.615.", "meta": {"id": "WUA_STAT_25K_012197", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.37, s=28.18, n=81, compute CI."}]}, "response": "95% CI ≈ [71.23, 83.51].", "meta": {"id": "WUA_STAT_25K_012198", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=155.72, s=23.1, n=16, compute CI."}]}, "response": "95% CI ≈ [144.40, 167.04].", "meta": {"id": "WUA_STAT_25K_012199", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.56, s=35.36, n=195, compute CI."}]}, "response": "95% CI ≈ [162.60, 172.52].", "meta": {"id": "WUA_STAT_25K_012200", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=104.57, s=35.73, n=105, compute CI."}]}, "response": "95% CI ≈ [97.74, 111.40].", "meta": {"id": "WUA_STAT_25K_012201", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.46, μ=63.28, σ=22.22, compute z."}]}, "response": "z = -2.602.", "meta": {"id": "WUA_STAT_25K_012202", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.39, compute E and Var."}]}, "response": "E[X]=10.92; Var[X]=6.661.", "meta": {"id": "WUA_STAT_25K_012203", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.83, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.190.", "meta": {"id": "WUA_STAT_25K_012204", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.19, compute E and Var."}]}, "response": "E[X]=6.08; Var[X]=4.925.", "meta": {"id": "WUA_STAT_25K_012205", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.41, s=2.36, n=25, compute CI."}]}, "response": "95% CI ≈ [181.48, 183.34].", "meta": {"id": "WUA_STAT_25K_012206", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.82, s=33.83, n=10, compute CI."}]}, "response": "95% CI ≈ [38.85, 80.79].", "meta": {"id": "WUA_STAT_25K_012207", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.71, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.815.", "meta": {"id": "WUA_STAT_25K_012208", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.74, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.463.", "meta": {"id": "WUA_STAT_25K_012209", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=81.75, s=31.48, n=125, compute CI."}]}, "response": "95% CI ≈ [76.23, 87.27].", "meta": {"id": "WUA_STAT_25K_012210", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.26, μ=45.96, σ=19.47, compute z."}]}, "response": "z = -2.193.", "meta": {"id": "WUA_STAT_25K_012211", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.93, μ=8.09, σ=15.02, compute z."}]}, "response": "z = 5.981.", "meta": {"id": "WUA_STAT_25K_012212", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.21, μ=58.95, σ=23.18, compute z."}]}, "response": "z = -2.146.", "meta": {"id": "WUA_STAT_25K_012213", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.11, compute E and Var."}]}, "response": "E[X]=2.20; Var[X]=1.958.", "meta": {"id": "WUA_STAT_25K_012214", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.64, compute E and Var."}]}, "response": "E[X]=21.12; Var[X]=7.603.", "meta": {"id": "WUA_STAT_25K_012215", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.08, μ=74.88, σ=4.8, compute z."}]}, "response": "z = 3.167.", "meta": {"id": "WUA_STAT_25K_012216", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.12, μ=19.57, σ=14.22, compute z."}]}, "response": "z = 0.953.", "meta": {"id": "WUA_STAT_25K_012217", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.121, sens=0.82, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.287.", "meta": {"id": "WUA_STAT_25K_012218", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.47, compute E and Var."}]}, "response": "E[X]=9.40; Var[X]=4.982.", "meta": {"id": "WUA_STAT_25K_012219", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.71, μ=64.41, σ=16.17, compute z."}]}, "response": "z = -0.414.", "meta": {"id": "WUA_STAT_25K_012220", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.82, μ=3.31, σ=22.24, compute z."}]}, "response": "z = 3.036.", "meta": {"id": "WUA_STAT_25K_012221", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.89, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.047.", "meta": {"id": "WUA_STAT_25K_012222", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.43, compute E and Var."}]}, "response": "E[X]=11.18; Var[X]=6.373.", "meta": {"id": "WUA_STAT_25K_012223", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.5, compute E and Var."}]}, "response": "E[X]=12.00; Var[X]=6.000.", "meta": {"id": "WUA_STAT_25K_012224", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.21, μ=32.0, σ=4.11, compute z."}]}, "response": "z = -4.328.", "meta": {"id": "WUA_STAT_25K_012225", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.56, compute E and Var."}]}, "response": "E[X]=2.80; Var[X]=1.232.", "meta": {"id": "WUA_STAT_25K_012226", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.1, s=22.94, n=193, compute CI."}]}, "response": "95% CI ≈ [163.86, 170.34].", "meta": {"id": "WUA_STAT_25K_012227", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.22, s=14.18, n=146, compute CI."}]}, "response": "95% CI ≈ [41.92, 46.52].", "meta": {"id": "WUA_STAT_25K_012228", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.96, μ=35.26, σ=5.4, compute z."}]}, "response": "z = 9.019.", "meta": {"id": "WUA_STAT_25K_012229", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=141.76, s=2.02, n=40, compute CI."}]}, "response": "95% CI ≈ [141.13, 142.39].", "meta": {"id": "WUA_STAT_25K_012230", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.203, sens=0.78, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.713.", "meta": {"id": "WUA_STAT_25K_012231", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.4, μ=94.03, σ=16.48, compute z."}]}, "response": "z = -0.038.", "meta": {"id": "WUA_STAT_25K_012232", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.55, s=13.99, n=83, compute CI."}]}, "response": "95% CI ≈ [18.54, 24.56].", "meta": {"id": "WUA_STAT_25K_012233", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.56, compute E and Var."}]}, "response": "E[X]=11.20; Var[X]=4.928.", "meta": {"id": "WUA_STAT_25K_012234", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.3, μ=81.48, σ=5.02, compute z."}]}, "response": "z = 2.952.", "meta": {"id": "WUA_STAT_25K_012235", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.91, μ=32.48, σ=6.92, compute z."}]}, "response": "z = 1.796.", "meta": {"id": "WUA_STAT_25K_012236", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.16, compute E and Var."}]}, "response": "E[X]=1.44; Var[X]=1.210.", "meta": {"id": "WUA_STAT_25K_012237", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.29, compute E and Var."}]}, "response": "E[X]=6.67; Var[X]=4.736.", "meta": {"id": "WUA_STAT_25K_012238", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.72, μ=34.43, σ=12.14, compute z."}]}, "response": "z = 1.095.", "meta": {"id": "WUA_STAT_25K_012239", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.63, μ=18.12, σ=12.51, compute z."}]}, "response": "z = 4.277.", "meta": {"id": "WUA_STAT_25K_012240", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.86, μ=7.32, σ=11.36, compute z."}]}, "response": "z = 5.065.", "meta": {"id": "WUA_STAT_25K_012241", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=108.9, s=32.85, n=60, compute CI."}]}, "response": "95% CI ≈ [100.59, 117.21].", "meta": {"id": "WUA_STAT_25K_012242", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.67, μ=94.27, σ=10.77, compute z."}]}, "response": "z = -1.077.", "meta": {"id": "WUA_STAT_25K_012243", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.87, μ=59.15, σ=8.44, compute z."}]}, "response": "z = -3.469.", "meta": {"id": "WUA_STAT_25K_012244", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.87, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.598.", "meta": {"id": "WUA_STAT_25K_012245", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.6, μ=49.56, σ=10.02, compute z."}]}, "response": "z = -0.395.", "meta": {"id": "WUA_STAT_25K_012246", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.21, compute E and Var."}]}, "response": "E[X]=3.57; Var[X]=2.820.", "meta": {"id": "WUA_STAT_25K_012247", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.033, sens=0.87, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.106.", "meta": {"id": "WUA_STAT_25K_012248", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.66, s=1.03, n=82, compute CI."}]}, "response": "95% CI ≈ [172.44, 172.88].", "meta": {"id": "WUA_STAT_25K_012249", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.95, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.052.", "meta": {"id": "WUA_STAT_25K_012250", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.58, μ=23.69, σ=6.97, compute z."}]}, "response": "z = 9.884.", "meta": {"id": "WUA_STAT_25K_012251", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.038, sens=0.97, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.215.", "meta": {"id": "WUA_STAT_25K_012252", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.22, compute E and Var."}]}, "response": "E[X]=1.54; Var[X]=1.201.", "meta": {"id": "WUA_STAT_25K_012253", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.64, μ=10.55, σ=24.12, compute z."}]}, "response": "z = 1.372.", "meta": {"id": "WUA_STAT_25K_012254", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.55, s=35.71, n=38, compute CI."}]}, "response": "95% CI ≈ [120.20, 142.90].", "meta": {"id": "WUA_STAT_25K_012255", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.07, sens=0.73, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.256.", "meta": {"id": "WUA_STAT_25K_012256", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.26, compute E and Var."}]}, "response": "E[X]=9.36; Var[X]=6.926.", "meta": {"id": "WUA_STAT_25K_012257", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.92, s=22.57, n=61, compute CI."}]}, "response": "95% CI ≈ [7.26, 18.58].", "meta": {"id": "WUA_STAT_25K_012258", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.42, compute E and Var."}]}, "response": "E[X]=4.20; Var[X]=2.436.", "meta": {"id": "WUA_STAT_25K_012259", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.67, μ=96.98, σ=14.52, compute z."}]}, "response": "z = -5.324.", "meta": {"id": "WUA_STAT_25K_012260", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.95, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.597.", "meta": {"id": "WUA_STAT_25K_012261", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.011, sens=0.94, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.052.", "meta": {"id": "WUA_STAT_25K_012262", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.02, sens=0.8, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.072.", "meta": {"id": "WUA_STAT_25K_012263", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.201, sens=0.78, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.536.", "meta": {"id": "WUA_STAT_25K_012264", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.42, μ=10.78, σ=2.37, compute z."}]}, "response": "z = 8.287.", "meta": {"id": "WUA_STAT_25K_012265", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.21, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=6.636.", "meta": {"id": "WUA_STAT_25K_012266", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.14, compute E and Var."}]}, "response": "E[X]=2.66; Var[X]=2.288.", "meta": {"id": "WUA_STAT_25K_012267", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.031, sens=0.71, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.362.", "meta": {"id": "WUA_STAT_25K_012268", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.89, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.774.", "meta": {"id": "WUA_STAT_25K_012269", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.141, sens=0.73, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.444.", "meta": {"id": "WUA_STAT_25K_012270", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.73, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.462.", "meta": {"id": "WUA_STAT_25K_012271", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.41, compute E and Var."}]}, "response": "E[X]=15.58; Var[X]=9.192.", "meta": {"id": "WUA_STAT_25K_012272", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.08, compute E and Var."}]}, "response": "E[X]=1.04; Var[X]=0.957.", "meta": {"id": "WUA_STAT_25K_012273", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.51, s=30.58, n=158, compute CI."}]}, "response": "95% CI ≈ [17.74, 27.28].", "meta": {"id": "WUA_STAT_25K_012274", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.49, μ=24.02, σ=11.16, compute z."}]}, "response": "z = 3.268.", "meta": {"id": "WUA_STAT_25K_012275", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.69, compute E and Var."}]}, "response": "E[X]=7.59; Var[X]=2.353.", "meta": {"id": "WUA_STAT_25K_012276", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.034, sens=0.87, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.304.", "meta": {"id": "WUA_STAT_25K_012277", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.59, compute E and Var."}]}, "response": "E[X]=15.93; Var[X]=6.531.", "meta": {"id": "WUA_STAT_25K_012278", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.13, μ=97.56, σ=22.29, compute z."}]}, "response": "z = 0.026.", "meta": {"id": "WUA_STAT_25K_012279", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.47, s=1.72, n=27, compute CI."}]}, "response": "95% CI ≈ [109.82, 111.12].", "meta": {"id": "WUA_STAT_25K_012280", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.14, s=26.37, n=39, compute CI."}]}, "response": "95% CI ≈ [9.86, 26.42].", "meta": {"id": "WUA_STAT_25K_012281", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.98, μ=72.51, σ=13.83, compute z."}]}, "response": "z = -0.183.", "meta": {"id": "WUA_STAT_25K_012282", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.74, s=37.28, n=16, compute CI."}]}, "response": "95% CI ≈ [71.47, 108.01].", "meta": {"id": "WUA_STAT_25K_012283", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.14, s=32.49, n=77, compute CI."}]}, "response": "95% CI ≈ [113.88, 128.40].", "meta": {"id": "WUA_STAT_25K_012284", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.72, μ=56.9, σ=10.12, compute z."}]}, "response": "z = -2.192.", "meta": {"id": "WUA_STAT_25K_012285", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.21, μ=15.6, σ=24.58, compute z."}]}, "response": "z = -0.097.", "meta": {"id": "WUA_STAT_25K_012286", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.247, sens=0.73, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.774.", "meta": {"id": "WUA_STAT_25K_012287", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.57, μ=50.48, σ=20.18, compute z."}]}, "response": "z = 1.342.", "meta": {"id": "WUA_STAT_25K_012288", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.77, compute E and Var."}]}, "response": "E[X]=8.47; Var[X]=1.948.", "meta": {"id": "WUA_STAT_25K_012289", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.39, compute E and Var."}]}, "response": "E[X]=10.53; Var[X]=6.423.", "meta": {"id": "WUA_STAT_25K_012290", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.042, sens=0.84, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.479.", "meta": {"id": "WUA_STAT_25K_012291", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.84, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.870.", "meta": {"id": "WUA_STAT_25K_012292", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.95, μ=9.82, σ=1.96, compute z."}]}, "response": "z = -4.526.", "meta": {"id": "WUA_STAT_25K_012293", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.52, μ=86.1, σ=3.1, compute z."}]}, "response": "z = -16.316.", "meta": {"id": "WUA_STAT_25K_012294", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.249, sens=0.79, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.512.", "meta": {"id": "WUA_STAT_25K_012295", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.15, μ=18.53, σ=2.38, compute z."}]}, "response": "z = 9.924.", "meta": {"id": "WUA_STAT_25K_012296", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.89, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.853.", "meta": {"id": "WUA_STAT_25K_012297", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.233, sens=0.82, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.624.", "meta": {"id": "WUA_STAT_25K_012298", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.91, s=4.16, n=14, compute CI."}]}, "response": "95% CI ≈ [159.73, 164.09].", "meta": {"id": "WUA_STAT_25K_012299", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.02, μ=83.7, σ=2.85, compute z."}]}, "response": "z = -4.800.", "meta": {"id": "WUA_STAT_25K_012300", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.06, s=21.76, n=133, compute CI."}]}, "response": "95% CI ≈ [61.36, 68.76].", "meta": {"id": "WUA_STAT_25K_012301", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.141, sens=0.78, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.865.", "meta": {"id": "WUA_STAT_25K_012302", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.033, sens=0.76, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.115.", "meta": {"id": "WUA_STAT_25K_012303", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.88, s=6.24, n=48, compute CI."}]}, "response": "95% CI ≈ [160.11, 163.65].", "meta": {"id": "WUA_STAT_25K_012304", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.89, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.128.", "meta": {"id": "WUA_STAT_25K_012305", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.3, compute E and Var."}]}, "response": "E[X]=5.40; Var[X]=3.780.", "meta": {"id": "WUA_STAT_25K_012306", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.65, s=24.89, n=166, compute CI."}]}, "response": "95% CI ≈ [195.86, 203.44].", "meta": {"id": "WUA_STAT_25K_012307", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.53, compute E and Var."}]}, "response": "E[X]=8.48; Var[X]=3.986.", "meta": {"id": "WUA_STAT_25K_012308", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.48, compute E and Var."}]}, "response": "E[X]=9.60; Var[X]=4.992.", "meta": {"id": "WUA_STAT_25K_012309", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.15, μ=40.16, σ=15.15, compute z."}]}, "response": "z = -1.783.", "meta": {"id": "WUA_STAT_25K_012310", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.21, s=35.88, n=25, compute CI."}]}, "response": "95% CI ≈ [115.15, 143.27].", "meta": {"id": "WUA_STAT_25K_012311", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.02, μ=90.19, σ=19.96, compute z."}]}, "response": "z = -0.860.", "meta": {"id": "WUA_STAT_25K_012312", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.27, compute E and Var."}]}, "response": "E[X]=4.05; Var[X]=2.957.", "meta": {"id": "WUA_STAT_25K_012313", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.23, μ=56.44, σ=6.98, compute z."}]}, "response": "z = -0.890.", "meta": {"id": "WUA_STAT_25K_012314", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.82, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.464.", "meta": {"id": "WUA_STAT_25K_012315", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.34, compute E and Var."}]}, "response": "E[X]=7.48; Var[X]=4.937.", "meta": {"id": "WUA_STAT_25K_012316", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.12, s=7.33, n=131, compute CI."}]}, "response": "95% CI ≈ [142.86, 145.38].", "meta": {"id": "WUA_STAT_25K_012317", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.88, s=23.5, n=151, compute CI."}]}, "response": "95% CI ≈ [133.13, 140.63].", "meta": {"id": "WUA_STAT_25K_012318", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.33, compute E and Var."}]}, "response": "E[X]=6.93; Var[X]=4.643.", "meta": {"id": "WUA_STAT_25K_012319", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.8, s=10.27, n=30, compute CI."}]}, "response": "95% CI ≈ [9.12, 16.48].", "meta": {"id": "WUA_STAT_25K_012320", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.73, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.653.", "meta": {"id": "WUA_STAT_25K_012321", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.18, μ=97.55, σ=8.32, compute z."}]}, "response": "z = -6.294.", "meta": {"id": "WUA_STAT_25K_012322", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.91, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.052.", "meta": {"id": "WUA_STAT_25K_012323", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.6, s=30.25, n=149, compute CI."}]}, "response": "95% CI ≈ [114.74, 124.46].", "meta": {"id": "WUA_STAT_25K_012324", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=158.03, s=22.9, n=80, compute CI."}]}, "response": "95% CI ≈ [153.01, 163.05].", "meta": {"id": "WUA_STAT_25K_012325", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.67, compute E and Var."}]}, "response": "E[X]=16.75; Var[X]=5.527.", "meta": {"id": "WUA_STAT_25K_012326", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.44, compute E and Var."}]}, "response": "E[X]=7.04; Var[X]=3.942.", "meta": {"id": "WUA_STAT_25K_012327", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.39, μ=47.58, σ=17.27, compute z."}]}, "response": "z = -2.038.", "meta": {"id": "WUA_STAT_25K_012328", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.3, compute E and Var."}]}, "response": "E[X]=1.50; Var[X]=1.050.", "meta": {"id": "WUA_STAT_25K_012329", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.9, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.280.", "meta": {"id": "WUA_STAT_25K_012330", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.8, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.716.", "meta": {"id": "WUA_STAT_25K_012331", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.096, sens=0.76, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.473.", "meta": {"id": "WUA_STAT_25K_012332", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.31, compute E and Var."}]}, "response": "E[X]=9.30; Var[X]=6.417.", "meta": {"id": "WUA_STAT_25K_012333", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.04, μ=77.98, σ=14.29, compute z."}]}, "response": "z = -3.285.", "meta": {"id": "WUA_STAT_25K_012334", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.12, s=28.28, n=93, compute CI."}]}, "response": "95% CI ≈ [18.37, 29.87].", "meta": {"id": "WUA_STAT_25K_012335", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.43, μ=87.17, σ=21.22, compute z."}]}, "response": "z = -0.930.", "meta": {"id": "WUA_STAT_25K_012336", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.111, sens=0.78, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.428.", "meta": {"id": "WUA_STAT_25K_012337", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.3, compute E and Var."}]}, "response": "E[X]=10.80; Var[X]=7.560.", "meta": {"id": "WUA_STAT_25K_012338", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.35, μ=14.39, σ=15.5, compute z."}]}, "response": "z = 3.546.", "meta": {"id": "WUA_STAT_25K_012339", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.09, compute E and Var."}]}, "response": "E[X]=2.07; Var[X]=1.884.", "meta": {"id": "WUA_STAT_25K_012340", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.61, μ=6.68, σ=22.69, compute z."}]}, "response": "z = 1.716.", "meta": {"id": "WUA_STAT_25K_012341", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.93, s=15.83, n=171, compute CI."}]}, "response": "95% CI ≈ [35.56, 40.30].", "meta": {"id": "WUA_STAT_25K_012342", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.62, μ=20.82, σ=15.34, compute z."}]}, "response": "z = 3.442.", "meta": {"id": "WUA_STAT_25K_012343", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.61, μ=60.99, σ=5.88, compute z."}]}, "response": "z = 2.997.", "meta": {"id": "WUA_STAT_25K_012344", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.27, compute E and Var."}]}, "response": "E[X]=2.43; Var[X]=1.774.", "meta": {"id": "WUA_STAT_25K_012345", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.2, s=11.73, n=125, compute CI."}]}, "response": "95% CI ≈ [46.14, 50.26].", "meta": {"id": "WUA_STAT_25K_012346", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.09, compute E and Var."}]}, "response": "E[X]=0.54; Var[X]=0.491.", "meta": {"id": "WUA_STAT_25K_012347", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.16, compute E and Var."}]}, "response": "E[X]=1.44; Var[X]=1.210.", "meta": {"id": "WUA_STAT_25K_012348", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.75, s=16.55, n=195, compute CI."}]}, "response": "95% CI ≈ [82.43, 87.07].", "meta": {"id": "WUA_STAT_25K_012349", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.66, compute E and Var."}]}, "response": "E[X]=19.14; Var[X]=6.508.", "meta": {"id": "WUA_STAT_25K_012350", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.81, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.796.", "meta": {"id": "WUA_STAT_25K_012351", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.27, μ=7.4, σ=19.02, compute z."}]}, "response": "z = 1.570.", "meta": {"id": "WUA_STAT_25K_012352", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.79, compute E and Var."}]}, "response": "E[X]=5.53; Var[X]=1.161.", "meta": {"id": "WUA_STAT_25K_012353", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.1, μ=99.07, σ=3.01, compute z."}]}, "response": "z = -28.894.", "meta": {"id": "WUA_STAT_25K_012354", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.122, sens=0.76, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.315.", "meta": {"id": "WUA_STAT_25K_012355", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.78, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.573.", "meta": {"id": "WUA_STAT_25K_012356", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.79, compute E and Var."}]}, "response": "E[X]=25.28; Var[X]=5.309.", "meta": {"id": "WUA_STAT_25K_012357", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.76, compute E and Var."}]}, "response": "E[X]=5.32; Var[X]=1.277.", "meta": {"id": "WUA_STAT_25K_012358", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.33, s=3.59, n=176, compute CI."}]}, "response": "95% CI ≈ [23.80, 24.86].", "meta": {"id": "WUA_STAT_25K_012359", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.75, μ=84.67, σ=21.15, compute z."}]}, "response": "z = -1.320.", "meta": {"id": "WUA_STAT_25K_012360", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.91, μ=90.43, σ=17.09, compute z."}]}, "response": "z = -2.254.", "meta": {"id": "WUA_STAT_25K_012361", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.8, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.804.", "meta": {"id": "WUA_STAT_25K_012362", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.66, s=34.01, n=165, compute CI."}]}, "response": "95% CI ≈ [73.47, 83.85].", "meta": {"id": "WUA_STAT_25K_012363", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.209, sens=0.74, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.429.", "meta": {"id": "WUA_STAT_25K_012364", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.134, sens=0.82, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.356.", "meta": {"id": "WUA_STAT_25K_012365", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.77, compute E and Var."}]}, "response": "E[X]=30.80; Var[X]=7.084.", "meta": {"id": "WUA_STAT_25K_012366", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.19, compute E and Var."}]}, "response": "E[X]=4.94; Var[X]=4.001.", "meta": {"id": "WUA_STAT_25K_012367", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.67, compute E and Var."}]}, "response": "E[X]=26.13; Var[X]=8.623.", "meta": {"id": "WUA_STAT_25K_012368", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.055, sens=0.94, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.354.", "meta": {"id": "WUA_STAT_25K_012369", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.23, compute E and Var."}]}, "response": "E[X]=1.84; Var[X]=1.417.", "meta": {"id": "WUA_STAT_25K_012370", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.45, μ=13.91, σ=8.25, compute z."}]}, "response": "z = -0.298.", "meta": {"id": "WUA_STAT_25K_012371", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.51, compute E and Var."}]}, "response": "E[X]=9.69; Var[X]=4.748.", "meta": {"id": "WUA_STAT_25K_012372", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.57, s=27.47, n=55, compute CI."}]}, "response": "95% CI ≈ [120.31, 134.83].", "meta": {"id": "WUA_STAT_25K_012373", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.64, s=30.05, n=167, compute CI."}]}, "response": "95% CI ≈ [32.08, 41.20].", "meta": {"id": "WUA_STAT_25K_012374", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.124, sens=0.97, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.632.", "meta": {"id": "WUA_STAT_25K_012375", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.75, s=39.53, n=136, compute CI."}]}, "response": "95% CI ≈ [154.11, 167.39].", "meta": {"id": "WUA_STAT_25K_012376", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.0, μ=40.06, σ=22.6, compute z."}]}, "response": "z = -1.640.", "meta": {"id": "WUA_STAT_25K_012377", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.5, compute E and Var."}]}, "response": "E[X]=5.00; Var[X]=2.500.", "meta": {"id": "WUA_STAT_25K_012378", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=68.99, s=32.17, n=74, compute CI."}]}, "response": "95% CI ≈ [61.66, 76.32].", "meta": {"id": "WUA_STAT_25K_012379", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.39, s=15.74, n=44, compute CI."}]}, "response": "95% CI ≈ [174.74, 184.04].", "meta": {"id": "WUA_STAT_25K_012380", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.96, μ=33.12, σ=17.39, compute z."}]}, "response": "z = 3.671.", "meta": {"id": "WUA_STAT_25K_012381", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.77, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.424.", "meta": {"id": "WUA_STAT_25K_012382", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.25, compute E and Var."}]}, "response": "E[X]=9.50; Var[X]=7.125.", "meta": {"id": "WUA_STAT_25K_012383", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=148.46, s=26.3, n=144, compute CI."}]}, "response": "95% CI ≈ [144.16, 152.76].", "meta": {"id": "WUA_STAT_25K_012384", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.06, μ=95.83, σ=20.66, compute z."}]}, "response": "z = -0.231.", "meta": {"id": "WUA_STAT_25K_012385", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=47.89, s=16.82, n=184, compute CI."}]}, "response": "95% CI ≈ [45.46, 50.32].", "meta": {"id": "WUA_STAT_25K_012386", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.6, compute E and Var."}]}, "response": "E[X]=5.40; Var[X]=2.160.", "meta": {"id": "WUA_STAT_25K_012387", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.73, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.603.", "meta": {"id": "WUA_STAT_25K_012388", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.42, s=18.51, n=122, compute CI."}]}, "response": "95% CI ≈ [26.14, 32.70].", "meta": {"id": "WUA_STAT_25K_012389", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.084, sens=0.89, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.386.", "meta": {"id": "WUA_STAT_25K_012390", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.13, compute E and Var."}]}, "response": "E[X]=3.51; Var[X]=3.054.", "meta": {"id": "WUA_STAT_25K_012391", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.42, compute E and Var."}]}, "response": "E[X]=3.36; Var[X]=1.949.", "meta": {"id": "WUA_STAT_25K_012392", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.15, μ=39.82, σ=17.63, compute z."}]}, "response": "z = -0.038.", "meta": {"id": "WUA_STAT_25K_012393", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=52.13, s=29.71, n=37, compute CI."}]}, "response": "95% CI ≈ [42.56, 61.70].", "meta": {"id": "WUA_STAT_25K_012394", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.46, μ=15.14, σ=24.0, compute z."}]}, "response": "z = -0.237.", "meta": {"id": "WUA_STAT_25K_012395", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.04, μ=21.76, σ=17.3, compute z."}]}, "response": "z = 2.849.", "meta": {"id": "WUA_STAT_25K_012396", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.08, μ=96.49, σ=4.3, compute z."}]}, "response": "z = -9.863.", "meta": {"id": "WUA_STAT_25K_012397", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.28, μ=23.07, σ=5.33, compute z."}]}, "response": "z = 10.734.", "meta": {"id": "WUA_STAT_25K_012398", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.059, sens=0.71, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.817.", "meta": {"id": "WUA_STAT_25K_012399", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.33, μ=14.52, σ=9.17, compute z."}]}, "response": "z = 5.105.", "meta": {"id": "WUA_STAT_25K_012400", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.059, sens=0.71, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.137.", "meta": {"id": "WUA_STAT_25K_012401", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=195.53, s=3.21, n=21, compute CI."}]}, "response": "95% CI ≈ [194.16, 196.90].", "meta": {"id": "WUA_STAT_25K_012402", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.054, sens=0.83, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.149.", "meta": {"id": "WUA_STAT_25K_012403", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.203, sens=0.8, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.911.", "meta": {"id": "WUA_STAT_25K_012404", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.48, μ=85.22, σ=12.67, compute z."}]}, "response": "z = -4.952.", "meta": {"id": "WUA_STAT_25K_012405", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.98, μ=72.91, σ=19.92, compute z."}]}, "response": "z = -2.808.", "meta": {"id": "WUA_STAT_25K_012406", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.87, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.386.", "meta": {"id": "WUA_STAT_25K_012407", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.28, compute E and Var."}]}, "response": "E[X]=7.00; Var[X]=5.040.", "meta": {"id": "WUA_STAT_25K_012408", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.73, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.317.", "meta": {"id": "WUA_STAT_25K_012409", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.63, compute E and Var."}]}, "response": "E[X]=17.64; Var[X]=6.527.", "meta": {"id": "WUA_STAT_25K_012410", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.88, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.804.", "meta": {"id": "WUA_STAT_25K_012411", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.21, s=29.46, n=157, compute CI."}]}, "response": "95% CI ≈ [189.60, 198.82].", "meta": {"id": "WUA_STAT_25K_012412", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.02, s=26.75, n=146, compute CI."}]}, "response": "95% CI ≈ [118.68, 127.36].", "meta": {"id": "WUA_STAT_25K_012413", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.93, μ=98.68, σ=2.65, compute z."}]}, "response": "z = -29.717.", "meta": {"id": "WUA_STAT_25K_012414", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.72, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.040.", "meta": {"id": "WUA_STAT_25K_012415", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.41, s=30.87, n=176, compute CI."}]}, "response": "95% CI ≈ [162.85, 171.97].", "meta": {"id": "WUA_STAT_25K_012416", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.34, s=10.48, n=158, compute CI."}]}, "response": "95% CI ≈ [22.71, 25.97].", "meta": {"id": "WUA_STAT_25K_012417", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.85, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.711.", "meta": {"id": "WUA_STAT_25K_012418", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.211, sens=0.87, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.492.", "meta": {"id": "WUA_STAT_25K_012419", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.9, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.515.", "meta": {"id": "WUA_STAT_25K_012420", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.95, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.479.", "meta": {"id": "WUA_STAT_25K_012421", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.92, s=31.03, n=168, compute CI."}]}, "response": "95% CI ≈ [162.23, 171.61].", "meta": {"id": "WUA_STAT_25K_012422", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.43, compute E and Var."}]}, "response": "E[X]=16.77; Var[X]=9.559.", "meta": {"id": "WUA_STAT_25K_012423", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.49, μ=23.45, σ=6.2, compute z."}]}, "response": "z = 10.329.", "meta": {"id": "WUA_STAT_25K_012424", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.46, compute E and Var."}]}, "response": "E[X]=15.18; Var[X]=8.197.", "meta": {"id": "WUA_STAT_25K_012425", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.32, μ=76.94, σ=4.68, compute z."}]}, "response": "z = -6.756.", "meta": {"id": "WUA_STAT_25K_012426", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.03, μ=88.32, σ=10.76, compute z."}]}, "response": "z = -2.443.", "meta": {"id": "WUA_STAT_25K_012427", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.16, compute E and Var."}]}, "response": "E[X]=4.00; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_012428", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.75, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.830.", "meta": {"id": "WUA_STAT_25K_012429", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.143, sens=0.94, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.887.", "meta": {"id": "WUA_STAT_25K_012430", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.78, compute E and Var."}]}, "response": "E[X]=6.24; Var[X]=1.373.", "meta": {"id": "WUA_STAT_25K_012431", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.41, compute E and Var."}]}, "response": "E[X]=16.40; Var[X]=9.676.", "meta": {"id": "WUA_STAT_25K_012432", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.65, compute E and Var."}]}, "response": "E[X]=5.20; Var[X]=1.820.", "meta": {"id": "WUA_STAT_25K_012433", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.94, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.956.", "meta": {"id": "WUA_STAT_25K_012434", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.34, s=39.96, n=171, compute CI."}]}, "response": "95% CI ≈ [38.35, 50.33].", "meta": {"id": "WUA_STAT_25K_012435", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.85, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.330.", "meta": {"id": "WUA_STAT_25K_012436", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.18, s=16.66, n=170, compute CI."}]}, "response": "95% CI ≈ [157.68, 162.68].", "meta": {"id": "WUA_STAT_25K_012437", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.57, compute E and Var."}]}, "response": "E[X]=11.40; Var[X]=4.902.", "meta": {"id": "WUA_STAT_25K_012438", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.049, sens=0.9, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.279.", "meta": {"id": "WUA_STAT_25K_012439", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.73, s=24.47, n=58, compute CI."}]}, "response": "95% CI ≈ [131.43, 144.03].", "meta": {"id": "WUA_STAT_25K_012440", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.175, sens=0.88, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.554.", "meta": {"id": "WUA_STAT_25K_012441", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.31, s=3.46, n=200, compute CI."}]}, "response": "95% CI ≈ [184.83, 185.79].", "meta": {"id": "WUA_STAT_25K_012442", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.22, compute E and Var."}]}, "response": "E[X]=2.64; Var[X]=2.059.", "meta": {"id": "WUA_STAT_25K_012443", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.91, μ=98.67, σ=8.28, compute z."}]}, "response": "z = -2.870.", "meta": {"id": "WUA_STAT_25K_012444", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.74, s=21.12, n=101, compute CI."}]}, "response": "95% CI ≈ [173.62, 181.86].", "meta": {"id": "WUA_STAT_25K_012445", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=164.2, s=35.89, n=137, compute CI."}]}, "response": "95% CI ≈ [158.19, 170.21].", "meta": {"id": "WUA_STAT_25K_012446", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.08, compute E and Var."}]}, "response": "E[X]=1.04; Var[X]=0.957.", "meta": {"id": "WUA_STAT_25K_012447", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.06, compute E and Var."}]}, "response": "E[X]=2.04; Var[X]=1.918.", "meta": {"id": "WUA_STAT_25K_012448", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.08, compute E and Var."}]}, "response": "E[X]=0.96; Var[X]=0.883.", "meta": {"id": "WUA_STAT_25K_012449", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.65, compute E and Var."}]}, "response": "E[X]=15.60; Var[X]=5.460.", "meta": {"id": "WUA_STAT_25K_012450", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.143, sens=0.94, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.635.", "meta": {"id": "WUA_STAT_25K_012451", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.31, s=17.54, n=101, compute CI."}]}, "response": "95% CI ≈ [85.89, 92.73].", "meta": {"id": "WUA_STAT_25K_012452", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.01, μ=19.34, σ=3.34, compute z."}]}, "response": "z = 0.799.", "meta": {"id": "WUA_STAT_25K_012453", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.3, compute E and Var."}]}, "response": "E[X]=5.10; Var[X]=3.570.", "meta": {"id": "WUA_STAT_25K_012454", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.91, s=24.63, n=183, compute CI."}]}, "response": "95% CI ≈ [195.34, 202.48].", "meta": {"id": "WUA_STAT_25K_012455", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.2, s=2.45, n=152, compute CI."}]}, "response": "95% CI ≈ [141.81, 142.59].", "meta": {"id": "WUA_STAT_25K_012456", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.9, s=37.64, n=139, compute CI."}]}, "response": "95% CI ≈ [193.64, 206.16].", "meta": {"id": "WUA_STAT_25K_012457", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.53, compute E and Var."}]}, "response": "E[X]=18.55; Var[X]=8.719.", "meta": {"id": "WUA_STAT_25K_012458", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.04, μ=32.07, σ=10.8, compute z."}]}, "response": "z = 3.979.", "meta": {"id": "WUA_STAT_25K_012459", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.5, μ=34.44, σ=22.6, compute z."}]}, "response": "z = 2.569.", "meta": {"id": "WUA_STAT_25K_012460", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.09, s=33.66, n=115, compute CI."}]}, "response": "95% CI ≈ [173.94, 186.24].", "meta": {"id": "WUA_STAT_25K_012461", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.25, compute E and Var."}]}, "response": "E[X]=6.25; Var[X]=4.688.", "meta": {"id": "WUA_STAT_25K_012462", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.06, compute E and Var."}]}, "response": "E[X]=0.90; Var[X]=0.846.", "meta": {"id": "WUA_STAT_25K_012463", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.99, μ=61.82, σ=7.07, compute z."}]}, "response": "z = -2.946.", "meta": {"id": "WUA_STAT_25K_012464", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.44, μ=91.61, σ=20.32, compute z."}]}, "response": "z = -0.796.", "meta": {"id": "WUA_STAT_25K_012465", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.207, sens=0.74, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.563.", "meta": {"id": "WUA_STAT_25K_012466", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.26, compute E and Var."}]}, "response": "E[X]=7.54; Var[X]=5.580.", "meta": {"id": "WUA_STAT_25K_012467", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.85, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.605.", "meta": {"id": "WUA_STAT_25K_012468", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.44, μ=97.92, σ=2.55, compute z."}]}, "response": "z = -4.894.", "meta": {"id": "WUA_STAT_25K_012469", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.76, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.531.", "meta": {"id": "WUA_STAT_25K_012470", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.91, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.671.", "meta": {"id": "WUA_STAT_25K_012471", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.88, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.201.", "meta": {"id": "WUA_STAT_25K_012472", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.38, s=10.18, n=187, compute CI."}]}, "response": "95% CI ≈ [43.92, 46.84].", "meta": {"id": "WUA_STAT_25K_012473", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.7, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.371.", "meta": {"id": "WUA_STAT_25K_012474", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.12, μ=50.5, σ=9.14, compute z."}]}, "response": "z = 2.256.", "meta": {"id": "WUA_STAT_25K_012475", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.16, s=3.63, n=169, compute CI."}]}, "response": "95% CI ≈ [82.61, 83.71].", "meta": {"id": "WUA_STAT_25K_012476", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.171, sens=0.85, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.412.", "meta": {"id": "WUA_STAT_25K_012477", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.18, sens=0.7, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.939.", "meta": {"id": "WUA_STAT_25K_012478", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.56, compute E and Var."}]}, "response": "E[X]=16.80; Var[X]=7.392.", "meta": {"id": "WUA_STAT_25K_012479", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.35, s=39.42, n=149, compute CI."}]}, "response": "95% CI ≈ [125.02, 137.68].", "meta": {"id": "WUA_STAT_25K_012480", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.229, sens=0.74, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.489.", "meta": {"id": "WUA_STAT_25K_012481", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.75, μ=51.15, σ=22.08, compute z."}]}, "response": "z = -2.101.", "meta": {"id": "WUA_STAT_25K_012482", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.82, μ=18.46, σ=23.12, compute z."}]}, "response": "z = 2.957.", "meta": {"id": "WUA_STAT_25K_012483", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.87, μ=60.18, σ=17.99, compute z."}]}, "response": "z = -3.074.", "meta": {"id": "WUA_STAT_25K_012484", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.75, s=9.15, n=51, compute CI."}]}, "response": "95% CI ≈ [134.24, 139.26].", "meta": {"id": "WUA_STAT_25K_012485", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.72, μ=54.48, σ=17.46, compute z."}]}, "response": "z = 0.357.", "meta": {"id": "WUA_STAT_25K_012486", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.62, compute E and Var."}]}, "response": "E[X]=3.72; Var[X]=1.414.", "meta": {"id": "WUA_STAT_25K_012487", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.21, compute E and Var."}]}, "response": "E[X]=4.62; Var[X]=3.650.", "meta": {"id": "WUA_STAT_25K_012488", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.61, μ=50.74, σ=19.61, compute z."}]}, "response": "z = 1.421.", "meta": {"id": "WUA_STAT_25K_012489", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.59, s=32.85, n=129, compute CI."}]}, "response": "95% CI ≈ [59.92, 71.26].", "meta": {"id": "WUA_STAT_25K_012490", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.03, μ=78.26, σ=14.91, compute z."}]}, "response": "z = 0.655.", "meta": {"id": "WUA_STAT_25K_012491", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.24, compute E and Var."}]}, "response": "E[X]=4.80; Var[X]=3.648.", "meta": {"id": "WUA_STAT_25K_012492", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.78, μ=51.61, σ=15.96, compute z."}]}, "response": "z = -1.932.", "meta": {"id": "WUA_STAT_25K_012493", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.89, μ=52.3, σ=19.2, compute z."}]}, "response": "z = -1.636.", "meta": {"id": "WUA_STAT_25K_012494", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.95, s=37.84, n=24, compute CI."}]}, "response": "95% CI ≈ [138.81, 169.09].", "meta": {"id": "WUA_STAT_25K_012495", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.36, s=38.12, n=160, compute CI."}]}, "response": "95% CI ≈ [24.45, 36.27].", "meta": {"id": "WUA_STAT_25K_012496", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.43, μ=26.46, σ=2.5, compute z."}]}, "response": "z = 2.788.", "meta": {"id": "WUA_STAT_25K_012497", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.98, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.504.", "meta": {"id": "WUA_STAT_25K_012498", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.76, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.479.", "meta": {"id": "WUA_STAT_25K_012499", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=81.82, s=36.57, n=174, compute CI."}]}, "response": "95% CI ≈ [76.39, 87.25].", "meta": {"id": "WUA_STAT_25K_012500", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.75, compute E and Var."}]}, "response": "E[X]=27.75; Var[X]=6.938.", "meta": {"id": "WUA_STAT_25K_012501", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.52, s=23.9, n=40, compute CI."}]}, "response": "95% CI ≈ [102.11, 116.93].", "meta": {"id": "WUA_STAT_25K_012502", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.7, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.290.", "meta": {"id": "WUA_STAT_25K_012503", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.222, sens=0.76, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.436.", "meta": {"id": "WUA_STAT_25K_012504", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.21, μ=47.68, σ=16.8, compute z."}]}, "response": "z = -1.397.", "meta": {"id": "WUA_STAT_25K_012505", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.133, sens=0.95, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.447.", "meta": {"id": "WUA_STAT_25K_012506", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.99, s=18.28, n=144, compute CI."}]}, "response": "95% CI ≈ [76.00, 81.98].", "meta": {"id": "WUA_STAT_25K_012507", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.42, μ=69.31, σ=21.37, compute z."}]}, "response": "z = 0.426.", "meta": {"id": "WUA_STAT_25K_012508", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.27, compute E and Var."}]}, "response": "E[X]=3.78; Var[X]=2.759.", "meta": {"id": "WUA_STAT_25K_012509", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.36, compute E and Var."}]}, "response": "E[X]=5.40; Var[X]=3.456.", "meta": {"id": "WUA_STAT_25K_012510", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.61, μ=46.21, σ=9.35, compute z."}]}, "response": "z = 1.647.", "meta": {"id": "WUA_STAT_25K_012511", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.72, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.681.", "meta": {"id": "WUA_STAT_25K_012512", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.36, compute E and Var."}]}, "response": "E[X]=7.56; Var[X]=4.838.", "meta": {"id": "WUA_STAT_25K_012513", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.44, compute E and Var."}]}, "response": "E[X]=10.56; Var[X]=5.914.", "meta": {"id": "WUA_STAT_25K_012514", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.2, s=38.35, n=86, compute CI."}]}, "response": "95% CI ≈ [25.09, 41.31].", "meta": {"id": "WUA_STAT_25K_012515", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.12, s=2.94, n=78, compute CI."}]}, "response": "95% CI ≈ [143.47, 144.77].", "meta": {"id": "WUA_STAT_25K_012516", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.53, s=7.65, n=158, compute CI."}]}, "response": "95% CI ≈ [113.34, 115.72].", "meta": {"id": "WUA_STAT_25K_012517", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.94, μ=65.51, σ=19.7, compute z."}]}, "response": "z = -1.196.", "meta": {"id": "WUA_STAT_25K_012518", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=158.11, s=32.72, n=192, compute CI."}]}, "response": "95% CI ≈ [153.48, 162.74].", "meta": {"id": "WUA_STAT_25K_012519", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.32, μ=49.43, σ=22.4, compute z."}]}, "response": "z = -1.210.", "meta": {"id": "WUA_STAT_25K_012520", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.07, compute E and Var."}]}, "response": "E[X]=0.84; Var[X]=0.781.", "meta": {"id": "WUA_STAT_25K_012521", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.36, compute E and Var."}]}, "response": "E[X]=6.12; Var[X]=3.917.", "meta": {"id": "WUA_STAT_25K_012522", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.33, compute E and Var."}]}, "response": "E[X]=11.22; Var[X]=7.517.", "meta": {"id": "WUA_STAT_25K_012523", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.121, sens=0.74, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.267.", "meta": {"id": "WUA_STAT_25K_012524", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.4, μ=28.29, σ=15.94, compute z."}]}, "response": "z = 2.955.", "meta": {"id": "WUA_STAT_25K_012525", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.222, sens=0.8, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.919.", "meta": {"id": "WUA_STAT_25K_012526", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.15, compute E and Var."}]}, "response": "E[X]=4.95; Var[X]=4.208.", "meta": {"id": "WUA_STAT_25K_012527", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.019, sens=0.74, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.115.", "meta": {"id": "WUA_STAT_25K_012528", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.2, μ=38.69, σ=19.14, compute z."}]}, "response": "z = 1.019.", "meta": {"id": "WUA_STAT_25K_012529", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.244, sens=0.82, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.546.", "meta": {"id": "WUA_STAT_25K_012530", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.68, compute E and Var."}]}, "response": "E[X]=23.12; Var[X]=7.398.", "meta": {"id": "WUA_STAT_25K_012531", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.23, compute E and Var."}]}, "response": "E[X]=2.30; Var[X]=1.771.", "meta": {"id": "WUA_STAT_25K_012532", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.22, compute E and Var."}]}, "response": "E[X]=1.54; Var[X]=1.201.", "meta": {"id": "WUA_STAT_25K_012533", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.18, sens=0.73, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.471.", "meta": {"id": "WUA_STAT_25K_012534", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.93, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.852.", "meta": {"id": "WUA_STAT_25K_012535", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.225, sens=0.77, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.568.", "meta": {"id": "WUA_STAT_25K_012536", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.78, compute E and Var."}]}, "response": "E[X]=4.68; Var[X]=1.030.", "meta": {"id": "WUA_STAT_25K_012537", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.69, compute E and Var."}]}, "response": "E[X]=15.18; Var[X]=4.706.", "meta": {"id": "WUA_STAT_25K_012538", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.8, μ=25.11, σ=6.99, compute z."}]}, "response": "z = 1.959.", "meta": {"id": "WUA_STAT_25K_012539", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.31, μ=86.39, σ=13.6, compute z."}]}, "response": "z = -2.065.", "meta": {"id": "WUA_STAT_25K_012540", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=19.28, s=23.23, n=133, compute CI."}]}, "response": "95% CI ≈ [15.33, 23.23].", "meta": {"id": "WUA_STAT_25K_012541", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.86, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.482.", "meta": {"id": "WUA_STAT_25K_012542", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.49, compute E and Var."}]}, "response": "E[X]=12.74; Var[X]=6.497.", "meta": {"id": "WUA_STAT_25K_012543", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.04, μ=9.11, σ=17.77, compute z."}]}, "response": "z = 2.866.", "meta": {"id": "WUA_STAT_25K_012544", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.65, μ=37.69, σ=5.53, compute z."}]}, "response": "z = 4.333.", "meta": {"id": "WUA_STAT_25K_012545", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.26, compute E and Var."}]}, "response": "E[X]=6.50; Var[X]=4.810.", "meta": {"id": "WUA_STAT_25K_012546", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.61, μ=29.8, σ=24.03, compute z."}]}, "response": "z = -0.382.", "meta": {"id": "WUA_STAT_25K_012547", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.06, compute E and Var."}]}, "response": "E[X]=1.62; Var[X]=1.523.", "meta": {"id": "WUA_STAT_25K_012548", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.64, s=11.58, n=81, compute CI."}]}, "response": "95% CI ≈ [16.12, 21.16].", "meta": {"id": "WUA_STAT_25K_012549", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.25, compute E and Var."}]}, "response": "E[X]=9.75; Var[X]=7.312.", "meta": {"id": "WUA_STAT_25K_012550", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.88, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.685.", "meta": {"id": "WUA_STAT_25K_012551", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.49, compute E and Var."}]}, "response": "E[X]=19.11; Var[X]=9.746.", "meta": {"id": "WUA_STAT_25K_012552", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.232, sens=0.87, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.745.", "meta": {"id": "WUA_STAT_25K_012553", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.15, compute E and Var."}]}, "response": "E[X]=5.55; Var[X]=4.717.", "meta": {"id": "WUA_STAT_25K_012554", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.15, compute E and Var."}]}, "response": "E[X]=0.75; Var[X]=0.637.", "meta": {"id": "WUA_STAT_25K_012555", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.38, s=11.32, n=133, compute CI."}]}, "response": "95% CI ≈ [140.46, 144.30].", "meta": {"id": "WUA_STAT_25K_012556", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.71, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.635.", "meta": {"id": "WUA_STAT_25K_012557", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.03, s=3.51, n=108, compute CI."}]}, "response": "95% CI ≈ [87.37, 88.69].", "meta": {"id": "WUA_STAT_25K_012558", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.72, s=7.98, n=88, compute CI."}]}, "response": "95% CI ≈ [59.05, 62.39].", "meta": {"id": "WUA_STAT_25K_012559", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.95, s=2.94, n=151, compute CI."}]}, "response": "95% CI ≈ [34.48, 35.42].", "meta": {"id": "WUA_STAT_25K_012560", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.22, compute E and Var."}]}, "response": "E[X]=5.50; Var[X]=4.290.", "meta": {"id": "WUA_STAT_25K_012561", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.71, s=16.15, n=29, compute CI."}]}, "response": "95% CI ≈ [78.83, 90.59].", "meta": {"id": "WUA_STAT_25K_012562", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.81, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.480.", "meta": {"id": "WUA_STAT_25K_012563", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=124.84, s=16.62, n=189, compute CI."}]}, "response": "95% CI ≈ [122.47, 127.21].", "meta": {"id": "WUA_STAT_25K_012564", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.61, compute E and Var."}]}, "response": "E[X]=15.25; Var[X]=5.947.", "meta": {"id": "WUA_STAT_25K_012565", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.96, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.354.", "meta": {"id": "WUA_STAT_25K_012566", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.072, sens=0.74, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.277.", "meta": {"id": "WUA_STAT_25K_012567", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.52, μ=44.82, σ=16.22, compute z."}]}, "response": "z = 1.215.", "meta": {"id": "WUA_STAT_25K_012568", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.049, sens=0.76, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.140.", "meta": {"id": "WUA_STAT_25K_012569", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.75, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.452.", "meta": {"id": "WUA_STAT_25K_012570", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.065, sens=0.98, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.405.", "meta": {"id": "WUA_STAT_25K_012571", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.74, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.505.", "meta": {"id": "WUA_STAT_25K_012572", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.2, μ=71.23, σ=21.86, compute z."}]}, "response": "z = -3.021.", "meta": {"id": "WUA_STAT_25K_012573", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.46, μ=22.07, σ=23.82, compute z."}]}, "response": "z = 1.570.", "meta": {"id": "WUA_STAT_25K_012574", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.27, s=30.4, n=131, compute CI."}]}, "response": "95% CI ≈ [161.06, 171.48].", "meta": {"id": "WUA_STAT_25K_012575", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.05, compute E and Var."}]}, "response": "E[X]=1.30; Var[X]=1.235.", "meta": {"id": "WUA_STAT_25K_012576", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.61, μ=79.92, σ=1.9, compute z."}]}, "response": "z = -25.426.", "meta": {"id": "WUA_STAT_25K_012577", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=143.08, s=17.27, n=88, compute CI."}]}, "response": "95% CI ≈ [139.47, 146.69].", "meta": {"id": "WUA_STAT_25K_012578", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.72, s=13.58, n=72, compute CI."}]}, "response": "95% CI ≈ [136.58, 142.86].", "meta": {"id": "WUA_STAT_25K_012579", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.47, s=21.27, n=42, compute CI."}]}, "response": "95% CI ≈ [108.04, 120.90].", "meta": {"id": "WUA_STAT_25K_012580", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.62, μ=68.4, σ=1.99, compute z."}]}, "response": "z = -5.417.", "meta": {"id": "WUA_STAT_25K_012581", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.5, compute E and Var."}]}, "response": "E[X]=17.50; Var[X]=8.750.", "meta": {"id": "WUA_STAT_25K_012582", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.44, compute E and Var."}]}, "response": "E[X]=2.20; Var[X]=1.232.", "meta": {"id": "WUA_STAT_25K_012583", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.08, μ=84.99, σ=19.86, compute z."}]}, "response": "z = -2.110.", "meta": {"id": "WUA_STAT_25K_012584", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.78, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.798.", "meta": {"id": "WUA_STAT_25K_012585", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.232, sens=0.83, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.926.", "meta": {"id": "WUA_STAT_25K_012586", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.091, sens=0.73, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.278.", "meta": {"id": "WUA_STAT_25K_012587", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.83, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.476.", "meta": {"id": "WUA_STAT_25K_012588", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.19, compute E and Var."}]}, "response": "E[X]=5.13; Var[X]=4.155.", "meta": {"id": "WUA_STAT_25K_012589", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.62, μ=1.66, σ=16.29, compute z."}]}, "response": "z = -0.002.", "meta": {"id": "WUA_STAT_25K_012590", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.23, s=4.75, n=38, compute CI."}]}, "response": "95% CI ≈ [148.72, 151.74].", "meta": {"id": "WUA_STAT_25K_012591", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.6, compute E and Var."}]}, "response": "E[X]=14.40; Var[X]=5.760.", "meta": {"id": "WUA_STAT_25K_012592", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.75, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.289.", "meta": {"id": "WUA_STAT_25K_012593", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.68, compute E and Var."}]}, "response": "E[X]=14.96; Var[X]=4.787.", "meta": {"id": "WUA_STAT_25K_012594", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.205, sens=0.83, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.641.", "meta": {"id": "WUA_STAT_25K_012595", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.88, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.529.", "meta": {"id": "WUA_STAT_25K_012596", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.56, compute E and Var."}]}, "response": "E[X]=6.72; Var[X]=2.957.", "meta": {"id": "WUA_STAT_25K_012597", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.52, compute E and Var."}]}, "response": "E[X]=7.80; Var[X]=3.744.", "meta": {"id": "WUA_STAT_25K_012598", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.04, s=37.95, n=79, compute CI."}]}, "response": "95% CI ≈ [152.67, 169.41].", "meta": {"id": "WUA_STAT_25K_012599", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.179, sens=0.95, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.522.", "meta": {"id": "WUA_STAT_25K_012600", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.75, μ=41.39, σ=21.29, compute z."}]}, "response": "z = 1.708.", "meta": {"id": "WUA_STAT_25K_012601", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=164.22, s=31.09, n=158, compute CI."}]}, "response": "95% CI ≈ [159.37, 169.07].", "meta": {"id": "WUA_STAT_25K_012602", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.69, compute E and Var."}]}, "response": "E[X]=7.59; Var[X]=2.353.", "meta": {"id": "WUA_STAT_25K_012603", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.92, μ=41.35, σ=11.06, compute z."}]}, "response": "z = -1.938.", "meta": {"id": "WUA_STAT_25K_012604", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.79, s=37.85, n=74, compute CI."}]}, "response": "95% CI ≈ [164.17, 181.41].", "meta": {"id": "WUA_STAT_25K_012605", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.09, μ=35.79, σ=11.38, compute z."}]}, "response": "z = 1.257.", "meta": {"id": "WUA_STAT_25K_012606", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.83, μ=82.41, σ=9.66, compute z."}]}, "response": "z = -0.060.", "meta": {"id": "WUA_STAT_25K_012607", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.09, compute E and Var."}]}, "response": "E[X]=1.80; Var[X]=1.638.", "meta": {"id": "WUA_STAT_25K_012608", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.14, μ=51.65, σ=20.62, compute z."}]}, "response": "z = -1.286.", "meta": {"id": "WUA_STAT_25K_012609", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.46, μ=12.52, σ=7.46, compute z."}]}, "response": "z = -1.483.", "meta": {"id": "WUA_STAT_25K_012610", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.86, μ=10.48, σ=15.39, compute z."}]}, "response": "z = 3.404.", "meta": {"id": "WUA_STAT_25K_012611", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.214, sens=0.89, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.634.", "meta": {"id": "WUA_STAT_25K_012612", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.08, compute E and Var."}]}, "response": "E[X]=3.20; Var[X]=2.944.", "meta": {"id": "WUA_STAT_25K_012613", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.74, compute E and Var."}]}, "response": "E[X]=26.64; Var[X]=6.926.", "meta": {"id": "WUA_STAT_25K_012614", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.131, sens=0.83, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.343.", "meta": {"id": "WUA_STAT_25K_012615", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.86, s=22.13, n=41, compute CI."}]}, "response": "95% CI ≈ [176.09, 189.63].", "meta": {"id": "WUA_STAT_25K_012616", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.67, compute E and Var."}]}, "response": "E[X]=24.79; Var[X]=8.181.", "meta": {"id": "WUA_STAT_25K_012617", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.065, sens=0.93, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.219.", "meta": {"id": "WUA_STAT_25K_012618", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.94, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.667.", "meta": {"id": "WUA_STAT_25K_012619", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.56, compute E and Var."}]}, "response": "E[X]=18.48; Var[X]=8.131.", "meta": {"id": "WUA_STAT_25K_012620", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.24, μ=66.89, σ=20.56, compute z."}]}, "response": "z = -2.755.", "meta": {"id": "WUA_STAT_25K_012621", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.41, s=23.16, n=103, compute CI."}]}, "response": "95% CI ≈ [115.94, 124.88].", "meta": {"id": "WUA_STAT_25K_012622", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.29, μ=84.36, σ=4.27, compute z."}]}, "response": "z = -3.061.", "meta": {"id": "WUA_STAT_25K_012623", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.76, μ=29.06, σ=11.17, compute z."}]}, "response": "z = -1.101.", "meta": {"id": "WUA_STAT_25K_012624", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.35, μ=98.14, σ=12.52, compute z."}]}, "response": "z = -3.577.", "meta": {"id": "WUA_STAT_25K_012625", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.6, compute E and Var."}]}, "response": "E[X]=13.80; Var[X]=5.520.", "meta": {"id": "WUA_STAT_25K_012626", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.95, s=13.43, n=76, compute CI."}]}, "response": "95% CI ≈ [21.93, 27.97].", "meta": {"id": "WUA_STAT_25K_012627", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.86, s=39.42, n=13, compute CI."}]}, "response": "95% CI ≈ [68.43, 111.29].", "meta": {"id": "WUA_STAT_25K_012628", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.24, μ=12.48, σ=12.03, compute z."}]}, "response": "z = 5.300.", "meta": {"id": "WUA_STAT_25K_012629", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.94, s=20.95, n=177, compute CI."}]}, "response": "95% CI ≈ [40.85, 47.03].", "meta": {"id": "WUA_STAT_25K_012630", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.162, sens=0.73, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.668.", "meta": {"id": "WUA_STAT_25K_012631", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.23, μ=66.92, σ=6.68, compute z."}]}, "response": "z = -1.451.", "meta": {"id": "WUA_STAT_25K_012632", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.75, compute E and Var."}]}, "response": "E[X]=28.50; Var[X]=7.125.", "meta": {"id": "WUA_STAT_25K_012633", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.027, sens=0.73, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.070.", "meta": {"id": "WUA_STAT_25K_012634", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.73, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.324.", "meta": {"id": "WUA_STAT_25K_012635", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.69, μ=62.67, σ=6.65, compute z."}]}, "response": "z = -2.704.", "meta": {"id": "WUA_STAT_25K_012636", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.05, compute E and Var."}]}, "response": "E[X]=1.55; Var[X]=1.472.", "meta": {"id": "WUA_STAT_25K_012637", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.82, s=21.67, n=83, compute CI."}]}, "response": "95% CI ≈ [51.16, 60.48].", "meta": {"id": "WUA_STAT_25K_012638", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.71, s=4.18, n=90, compute CI."}]}, "response": "95% CI ≈ [36.85, 38.57].", "meta": {"id": "WUA_STAT_25K_012639", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.83, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.441.", "meta": {"id": "WUA_STAT_25K_012640", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.75, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.407.", "meta": {"id": "WUA_STAT_25K_012641", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.87, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.315.", "meta": {"id": "WUA_STAT_25K_012642", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.64, μ=18.58, σ=9.42, compute z."}]}, "response": "z = 1.917.", "meta": {"id": "WUA_STAT_25K_012643", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.064, sens=0.73, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.416.", "meta": {"id": "WUA_STAT_25K_012644", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.28, compute E and Var."}]}, "response": "E[X]=1.40; Var[X]=1.008.", "meta": {"id": "WUA_STAT_25K_012645", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.237, sens=0.7, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.626.", "meta": {"id": "WUA_STAT_25K_012646", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.09, compute E and Var."}]}, "response": "E[X]=3.06; Var[X]=2.785.", "meta": {"id": "WUA_STAT_25K_012647", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.64, compute E and Var."}]}, "response": "E[X]=8.32; Var[X]=2.995.", "meta": {"id": "WUA_STAT_25K_012648", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.49, μ=84.83, σ=19.04, compute z."}]}, "response": "z = -1.593.", "meta": {"id": "WUA_STAT_25K_012649", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.119, sens=0.95, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.368.", "meta": {"id": "WUA_STAT_25K_012650", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.73, μ=47.78, σ=3.53, compute z."}]}, "response": "z = -4.263.", "meta": {"id": "WUA_STAT_25K_012651", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.42, compute E and Var."}]}, "response": "E[X]=5.88; Var[X]=3.410.", "meta": {"id": "WUA_STAT_25K_012652", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.43, s=17.86, n=134, compute CI."}]}, "response": "95% CI ≈ [124.41, 130.45].", "meta": {"id": "WUA_STAT_25K_012653", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.75, compute E and Var."}]}, "response": "E[X]=12.00; Var[X]=3.000.", "meta": {"id": "WUA_STAT_25K_012654", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.98, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.437.", "meta": {"id": "WUA_STAT_25K_012655", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.69, μ=40.27, σ=23.74, compute z."}]}, "response": "z = -0.361.", "meta": {"id": "WUA_STAT_25K_012656", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.81, μ=19.64, σ=8.55, compute z."}]}, "response": "z = 0.722.", "meta": {"id": "WUA_STAT_25K_012657", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.7, μ=39.57, σ=10.78, compute z."}]}, "response": "z = -3.513.", "meta": {"id": "WUA_STAT_25K_012658", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.11, μ=69.37, σ=23.73, compute z."}]}, "response": "z = -0.390.", "meta": {"id": "WUA_STAT_25K_012659", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.18, s=37.73, n=104, compute CI."}]}, "response": "95% CI ≈ [23.93, 38.43].", "meta": {"id": "WUA_STAT_25K_012660", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.85, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.827.", "meta": {"id": "WUA_STAT_25K_012661", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.104, sens=0.86, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.277.", "meta": {"id": "WUA_STAT_25K_012662", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.06, compute E and Var."}]}, "response": "E[X]=1.38; Var[X]=1.297.", "meta": {"id": "WUA_STAT_25K_012663", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.209, sens=0.96, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.572.", "meta": {"id": "WUA_STAT_25K_012664", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.28, compute E and Var."}]}, "response": "E[X]=10.08; Var[X]=7.258.", "meta": {"id": "WUA_STAT_25K_012665", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.21, sens=0.89, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.826.", "meta": {"id": "WUA_STAT_25K_012666", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.23, s=4.27, n=156, compute CI."}]}, "response": "95% CI ≈ [40.56, 41.90].", "meta": {"id": "WUA_STAT_25K_012667", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.17, μ=30.43, σ=20.92, compute z."}]}, "response": "z = 1.804.", "meta": {"id": "WUA_STAT_25K_012668", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.157, sens=0.82, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.488.", "meta": {"id": "WUA_STAT_25K_012669", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.86, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.447.", "meta": {"id": "WUA_STAT_25K_012670", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.12, compute E and Var."}]}, "response": "E[X]=3.12; Var[X]=2.746.", "meta": {"id": "WUA_STAT_25K_012671", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.35, s=15.27, n=159, compute CI."}]}, "response": "95% CI ≈ [191.98, 196.72].", "meta": {"id": "WUA_STAT_25K_012672", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.48, s=37.3, n=79, compute CI."}]}, "response": "95% CI ≈ [10.25, 26.71].", "meta": {"id": "WUA_STAT_25K_012673", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.8, μ=49.02, σ=22.16, compute z."}]}, "response": "z = -0.597.", "meta": {"id": "WUA_STAT_25K_012674", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.93, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.277.", "meta": {"id": "WUA_STAT_25K_012675", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.46, s=34.24, n=86, compute CI."}]}, "response": "95% CI ≈ [122.22, 136.70].", "meta": {"id": "WUA_STAT_25K_012676", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.15, μ=65.85, σ=12.63, compute z."}]}, "response": "z = 1.053.", "meta": {"id": "WUA_STAT_25K_012677", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.25, compute E and Var."}]}, "response": "E[X]=8.00; Var[X]=6.000.", "meta": {"id": "WUA_STAT_25K_012678", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.185, sens=0.93, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.657.", "meta": {"id": "WUA_STAT_25K_012679", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.62, μ=57.67, σ=13.53, compute z."}]}, "response": "z = -3.551.", "meta": {"id": "WUA_STAT_25K_012680", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.89, s=13.17, n=194, compute CI."}]}, "response": "95% CI ≈ [183.04, 186.74].", "meta": {"id": "WUA_STAT_25K_012681", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.058, sens=0.82, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.335.", "meta": {"id": "WUA_STAT_25K_012682", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.238, sens=0.74, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.852.", "meta": {"id": "WUA_STAT_25K_012683", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.19, compute E and Var."}]}, "response": "E[X]=6.65; Var[X]=5.387.", "meta": {"id": "WUA_STAT_25K_012684", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.94, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.345.", "meta": {"id": "WUA_STAT_25K_012685", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.16, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=4.704.", "meta": {"id": "WUA_STAT_25K_012686", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.48, s=9.51, n=42, compute CI."}]}, "response": "95% CI ≈ [172.60, 178.36].", "meta": {"id": "WUA_STAT_25K_012687", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.73, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.439.", "meta": {"id": "WUA_STAT_25K_012688", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.22, μ=6.88, σ=4.11, compute z."}]}, "response": "z = 7.139.", "meta": {"id": "WUA_STAT_25K_012689", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.19, μ=50.92, σ=16.6, compute z."}]}, "response": "z = -1.249.", "meta": {"id": "WUA_STAT_25K_012690", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.69, compute E and Var."}]}, "response": "E[X]=10.35; Var[X]=3.209.", "meta": {"id": "WUA_STAT_25K_012691", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.091, sens=0.73, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.448.", "meta": {"id": "WUA_STAT_25K_012692", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.46, s=31.44, n=59, compute CI."}]}, "response": "95% CI ≈ [109.44, 125.48].", "meta": {"id": "WUA_STAT_25K_012693", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.54, μ=42.26, σ=9.41, compute z."}]}, "response": "z = 4.812.", "meta": {"id": "WUA_STAT_25K_012694", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.68, compute E and Var."}]}, "response": "E[X]=15.64; Var[X]=5.005.", "meta": {"id": "WUA_STAT_25K_012695", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.94, s=24.19, n=49, compute CI."}]}, "response": "95% CI ≈ [8.17, 21.71].", "meta": {"id": "WUA_STAT_25K_012696", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.95, μ=90.84, σ=4.78, compute z."}]}, "response": "z = -11.065.", "meta": {"id": "WUA_STAT_25K_012697", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.73, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.557.", "meta": {"id": "WUA_STAT_25K_012698", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.84, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.435.", "meta": {"id": "WUA_STAT_25K_012699", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.13, s=39.86, n=47, compute CI."}]}, "response": "95% CI ≈ [178.73, 201.53].", "meta": {"id": "WUA_STAT_25K_012700", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.27, compute E and Var."}]}, "response": "E[X]=5.13; Var[X]=3.745.", "meta": {"id": "WUA_STAT_25K_012701", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.33, compute E and Var."}]}, "response": "E[X]=4.29; Var[X]=2.874.", "meta": {"id": "WUA_STAT_25K_012702", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.44, s=12.11, n=195, compute CI."}]}, "response": "95% CI ≈ [176.74, 180.14].", "meta": {"id": "WUA_STAT_25K_012703", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.16, μ=11.63, σ=10.9, compute z."}]}, "response": "z = 0.783.", "meta": {"id": "WUA_STAT_25K_012704", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.62, μ=64.29, σ=4.78, compute z."}]}, "response": "z = -5.579.", "meta": {"id": "WUA_STAT_25K_012705", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.034, sens=0.74, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.465.", "meta": {"id": "WUA_STAT_25K_012706", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.24, s=33.51, n=175, compute CI."}]}, "response": "95% CI ≈ [167.28, 177.20].", "meta": {"id": "WUA_STAT_25K_012707", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.72, compute E and Var."}]}, "response": "E[X]=10.80; Var[X]=3.024.", "meta": {"id": "WUA_STAT_25K_012708", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.29, s=20.39, n=103, compute CI."}]}, "response": "95% CI ≈ [27.35, 35.23].", "meta": {"id": "WUA_STAT_25K_012709", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.3, s=19.82, n=125, compute CI."}]}, "response": "95% CI ≈ [162.83, 169.77].", "meta": {"id": "WUA_STAT_25K_012710", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.87, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.079.", "meta": {"id": "WUA_STAT_25K_012711", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.96, s=14.75, n=180, compute CI."}]}, "response": "95% CI ≈ [155.81, 160.11].", "meta": {"id": "WUA_STAT_25K_012712", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.22, compute E and Var."}]}, "response": "E[X]=6.60; Var[X]=5.148.", "meta": {"id": "WUA_STAT_25K_012713", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.14, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=2.167.", "meta": {"id": "WUA_STAT_25K_012714", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.21, compute E and Var."}]}, "response": "E[X]=5.25; Var[X]=4.147.", "meta": {"id": "WUA_STAT_25K_012715", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.95, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.591.", "meta": {"id": "WUA_STAT_25K_012716", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.78, μ=61.64, σ=14.84, compute z."}]}, "response": "z = 0.616.", "meta": {"id": "WUA_STAT_25K_012717", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.02, μ=79.94, σ=8.75, compute z."}]}, "response": "z = -6.048.", "meta": {"id": "WUA_STAT_25K_012718", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.97, s=25.09, n=190, compute CI."}]}, "response": "95% CI ≈ [127.40, 134.54].", "meta": {"id": "WUA_STAT_25K_012719", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=149.99, s=16.31, n=100, compute CI."}]}, "response": "95% CI ≈ [146.79, 153.19].", "meta": {"id": "WUA_STAT_25K_012720", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.58, compute E and Var."}]}, "response": "E[X]=16.24; Var[X]=6.821.", "meta": {"id": "WUA_STAT_25K_012721", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=71.78, s=12.94, n=128, compute CI."}]}, "response": "95% CI ≈ [69.54, 74.02].", "meta": {"id": "WUA_STAT_25K_012722", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.16, compute E and Var."}]}, "response": "E[X]=2.24; Var[X]=1.882.", "meta": {"id": "WUA_STAT_25K_012723", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.89, μ=38.66, σ=12.71, compute z."}]}, "response": "z = -0.454.", "meta": {"id": "WUA_STAT_25K_012724", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.58, μ=42.92, σ=15.87, compute z."}]}, "response": "z = 3.507.", "meta": {"id": "WUA_STAT_25K_012725", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.07, μ=76.2, σ=21.36, compute z."}]}, "response": "z = -0.100.", "meta": {"id": "WUA_STAT_25K_012726", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=64.34, s=8.14, n=73, compute CI."}]}, "response": "95% CI ≈ [62.47, 66.21].", "meta": {"id": "WUA_STAT_25K_012727", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.12, μ=29.31, σ=20.0, compute z."}]}, "response": "z = 1.891.", "meta": {"id": "WUA_STAT_25K_012728", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.229, sens=0.82, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.575.", "meta": {"id": "WUA_STAT_25K_012729", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.158, sens=0.95, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.527.", "meta": {"id": "WUA_STAT_25K_012730", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.68, μ=23.95, σ=2.79, compute z."}]}, "response": "z = 26.427.", "meta": {"id": "WUA_STAT_25K_012731", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.46, μ=84.76, σ=6.19, compute z."}]}, "response": "z = 1.244.", "meta": {"id": "WUA_STAT_25K_012732", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.86, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.052.", "meta": {"id": "WUA_STAT_25K_012733", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.166, sens=0.94, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.418.", "meta": {"id": "WUA_STAT_25K_012734", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.17, s=2.04, n=156, compute CI."}]}, "response": "95% CI ≈ [137.85, 138.49].", "meta": {"id": "WUA_STAT_25K_012735", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.72, compute E and Var."}]}, "response": "E[X]=23.76; Var[X]=6.653.", "meta": {"id": "WUA_STAT_25K_012736", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.143, sens=0.97, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.698.", "meta": {"id": "WUA_STAT_25K_012737", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.135, sens=0.83, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.649.", "meta": {"id": "WUA_STAT_25K_012738", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.22, s=28.59, n=53, compute CI."}]}, "response": "95% CI ≈ [53.52, 68.92].", "meta": {"id": "WUA_STAT_25K_012739", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.93, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.560.", "meta": {"id": "WUA_STAT_25K_012740", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.71, compute E and Var."}]}, "response": "E[X]=4.26; Var[X]=1.235.", "meta": {"id": "WUA_STAT_25K_012741", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.75, s=29.41, n=151, compute CI."}]}, "response": "95% CI ≈ [155.06, 164.44].", "meta": {"id": "WUA_STAT_25K_012742", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.35, compute E and Var."}]}, "response": "E[X]=6.65; Var[X]=4.322.", "meta": {"id": "WUA_STAT_25K_012743", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.14, compute E and Var."}]}, "response": "E[X]=4.90; Var[X]=4.214.", "meta": {"id": "WUA_STAT_25K_012744", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.16, compute E and Var."}]}, "response": "E[X]=2.24; Var[X]=1.882.", "meta": {"id": "WUA_STAT_25K_012745", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.182, sens=0.78, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.505.", "meta": {"id": "WUA_STAT_25K_012746", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.56, s=2.15, n=73, compute CI."}]}, "response": "95% CI ≈ [170.07, 171.05].", "meta": {"id": "WUA_STAT_25K_012747", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.48, μ=94.68, σ=13.4, compute z."}]}, "response": "z = -5.313.", "meta": {"id": "WUA_STAT_25K_012748", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.86, μ=64.38, σ=6.08, compute z."}]}, "response": "z = -9.132.", "meta": {"id": "WUA_STAT_25K_012749", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.72, μ=92.17, σ=8.97, compute z."}]}, "response": "z = -3.952.", "meta": {"id": "WUA_STAT_25K_012750", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.91, μ=36.81, σ=7.8, compute z."}]}, "response": "z = -2.936.", "meta": {"id": "WUA_STAT_25K_012751", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.12, s=10.69, n=55, compute CI."}]}, "response": "95% CI ≈ [76.29, 81.95].", "meta": {"id": "WUA_STAT_25K_012752", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.062, sens=0.85, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.360.", "meta": {"id": "WUA_STAT_25K_012753", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.25, compute E and Var."}]}, "response": "E[X]=6.25; Var[X]=4.688.", "meta": {"id": "WUA_STAT_25K_012754", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.95, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.268.", "meta": {"id": "WUA_STAT_25K_012755", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.32, compute E and Var."}]}, "response": "E[X]=10.88; Var[X]=7.398.", "meta": {"id": "WUA_STAT_25K_012756", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=105.56, s=1.09, n=191, compute CI."}]}, "response": "95% CI ≈ [105.41, 105.71].", "meta": {"id": "WUA_STAT_25K_012757", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.06, μ=73.23, σ=18.83, compute z."}]}, "response": "z = -1.762.", "meta": {"id": "WUA_STAT_25K_012758", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.33, compute E and Var."}]}, "response": "E[X]=4.95; Var[X]=3.316.", "meta": {"id": "WUA_STAT_25K_012759", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.12, s=36.07, n=16, compute CI."}]}, "response": "95% CI ≈ [80.45, 115.79].", "meta": {"id": "WUA_STAT_25K_012760", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.61, s=13.58, n=72, compute CI."}]}, "response": "95% CI ≈ [183.47, 189.75].", "meta": {"id": "WUA_STAT_25K_012761", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.07, compute E and Var."}]}, "response": "E[X]=1.05; Var[X]=0.976.", "meta": {"id": "WUA_STAT_25K_012762", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.93, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.710.", "meta": {"id": "WUA_STAT_25K_012763", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.69, compute E and Var."}]}, "response": "E[X]=7.59; Var[X]=2.353.", "meta": {"id": "WUA_STAT_25K_012764", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.54, μ=50.32, σ=6.93, compute z."}]}, "response": "z = 6.958.", "meta": {"id": "WUA_STAT_25K_012765", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.18, s=6.37, n=17, compute CI."}]}, "response": "95% CI ≈ [45.15, 51.21].", "meta": {"id": "WUA_STAT_25K_012766", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.205, sens=0.9, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.563.", "meta": {"id": "WUA_STAT_25K_012767", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.5, μ=50.39, σ=13.98, compute z."}]}, "response": "z = -2.496.", "meta": {"id": "WUA_STAT_25K_012768", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.03, μ=5.39, σ=7.38, compute z."}]}, "response": "z = 1.713.", "meta": {"id": "WUA_STAT_25K_012769", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=191.15, s=23.57, n=196, compute CI."}]}, "response": "95% CI ≈ [187.85, 194.45].", "meta": {"id": "WUA_STAT_25K_012770", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.67, compute E and Var."}]}, "response": "E[X]=24.79; Var[X]=8.181.", "meta": {"id": "WUA_STAT_25K_012771", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.64, s=36.6, n=104, compute CI."}]}, "response": "95% CI ≈ [181.61, 195.67].", "meta": {"id": "WUA_STAT_25K_012772", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.89, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.245.", "meta": {"id": "WUA_STAT_25K_012773", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.08, compute E and Var."}]}, "response": "E[X]=1.20; Var[X]=1.104.", "meta": {"id": "WUA_STAT_25K_012774", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.98, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.446.", "meta": {"id": "WUA_STAT_25K_012775", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.1, compute E and Var."}]}, "response": "E[X]=3.10; Var[X]=2.790.", "meta": {"id": "WUA_STAT_25K_012776", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.92, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.730.", "meta": {"id": "WUA_STAT_25K_012777", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.23, sens=0.83, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.608.", "meta": {"id": "WUA_STAT_25K_012778", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.96, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.493.", "meta": {"id": "WUA_STAT_25K_012779", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.17, s=31.54, n=25, compute CI."}]}, "response": "95% CI ≈ [131.81, 156.53].", "meta": {"id": "WUA_STAT_25K_012780", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.27, μ=69.77, σ=3.16, compute z."}]}, "response": "z = -6.171.", "meta": {"id": "WUA_STAT_25K_012781", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.88, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.576.", "meta": {"id": "WUA_STAT_25K_012782", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.5, compute E and Var."}]}, "response": "E[X]=19.50; Var[X]=9.750.", "meta": {"id": "WUA_STAT_25K_012783", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.75, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.406.", "meta": {"id": "WUA_STAT_25K_012784", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.79, compute E and Var."}]}, "response": "E[X]=4.74; Var[X]=0.995.", "meta": {"id": "WUA_STAT_25K_012785", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.94, s=18.18, n=17, compute CI."}]}, "response": "95% CI ≈ [30.30, 47.58].", "meta": {"id": "WUA_STAT_25K_012786", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.034, sens=0.85, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.428.", "meta": {"id": "WUA_STAT_25K_012787", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.92, s=16.89, n=124, compute CI."}]}, "response": "95% CI ≈ [15.95, 21.89].", "meta": {"id": "WUA_STAT_25K_012788", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.047, sens=0.82, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.310.", "meta": {"id": "WUA_STAT_25K_012789", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.45, μ=24.96, σ=12.88, compute z."}]}, "response": "z = 0.038.", "meta": {"id": "WUA_STAT_25K_012790", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.28, μ=29.46, σ=1.43, compute z."}]}, "response": "z = 32.741.", "meta": {"id": "WUA_STAT_25K_012791", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.9, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.520.", "meta": {"id": "WUA_STAT_25K_012792", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.58, μ=41.57, σ=4.54, compute z."}]}, "response": "z = 11.015.", "meta": {"id": "WUA_STAT_25K_012793", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.87, s=36.27, n=45, compute CI."}]}, "response": "95% CI ≈ [128.27, 149.47].", "meta": {"id": "WUA_STAT_25K_012794", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=169.17, s=27.75, n=193, compute CI."}]}, "response": "95% CI ≈ [165.25, 173.09].", "meta": {"id": "WUA_STAT_25K_012795", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.13, μ=5.63, σ=5.29, compute z."}]}, "response": "z = 2.552.", "meta": {"id": "WUA_STAT_25K_012796", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.57, compute E and Var."}]}, "response": "E[X]=7.98; Var[X]=3.431.", "meta": {"id": "WUA_STAT_25K_012797", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.45, μ=52.88, σ=14.36, compute z."}]}, "response": "z = -1.283.", "meta": {"id": "WUA_STAT_25K_012798", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.78, compute E and Var."}]}, "response": "E[X]=9.36; Var[X]=2.059.", "meta": {"id": "WUA_STAT_25K_012799", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.01, s=19.43, n=83, compute CI."}]}, "response": "95% CI ≈ [116.83, 125.19].", "meta": {"id": "WUA_STAT_25K_012800", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.75, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.331.", "meta": {"id": "WUA_STAT_25K_012801", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.6, μ=37.53, σ=14.28, compute z."}]}, "response": "z = -1.396.", "meta": {"id": "WUA_STAT_25K_012802", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.28, μ=35.93, σ=3.97, compute z."}]}, "response": "z = 8.904.", "meta": {"id": "WUA_STAT_25K_012803", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.19, compute E and Var."}]}, "response": "E[X]=1.71; Var[X]=1.385.", "meta": {"id": "WUA_STAT_25K_012804", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.036, sens=0.93, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.537.", "meta": {"id": "WUA_STAT_25K_012805", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.95, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.370.", "meta": {"id": "WUA_STAT_25K_012806", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.49, s=12.67, n=22, compute CI."}]}, "response": "95% CI ≈ [74.20, 84.78].", "meta": {"id": "WUA_STAT_25K_012807", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.76, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.046.", "meta": {"id": "WUA_STAT_25K_012808", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.248, sens=0.74, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.589.", "meta": {"id": "WUA_STAT_25K_012809", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.84, μ=91.75, σ=18.41, compute z."}]}, "response": "z = -4.449.", "meta": {"id": "WUA_STAT_25K_012810", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.67, μ=83.8, σ=16.32, compute z."}]}, "response": "z = -1.417.", "meta": {"id": "WUA_STAT_25K_012811", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.74, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.784.", "meta": {"id": "WUA_STAT_25K_012812", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.73, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.205.", "meta": {"id": "WUA_STAT_25K_012813", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.82, μ=18.63, σ=15.52, compute z."}]}, "response": "z = 1.688.", "meta": {"id": "WUA_STAT_25K_012814", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.94, s=36.17, n=47, compute CI."}]}, "response": "95% CI ≈ [172.60, 193.28].", "meta": {"id": "WUA_STAT_25K_012815", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.69, compute E and Var."}]}, "response": "E[X]=10.35; Var[X]=3.209.", "meta": {"id": "WUA_STAT_25K_012816", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.135, sens=0.91, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.486.", "meta": {"id": "WUA_STAT_25K_012817", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.34, compute E and Var."}]}, "response": "E[X]=12.58; Var[X]=8.303.", "meta": {"id": "WUA_STAT_25K_012818", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.64, compute E and Var."}]}, "response": "E[X]=17.28; Var[X]=6.221.", "meta": {"id": "WUA_STAT_25K_012819", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.63, μ=33.9, σ=6.03, compute z."}]}, "response": "z = 3.438.", "meta": {"id": "WUA_STAT_25K_012820", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.3, μ=86.68, σ=4.18, compute z."}]}, "response": "z = 1.584.", "meta": {"id": "WUA_STAT_25K_012821", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.088, sens=0.81, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.465.", "meta": {"id": "WUA_STAT_25K_012822", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.238, sens=0.92, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.535.", "meta": {"id": "WUA_STAT_25K_012823", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.87, s=12.36, n=73, compute CI."}]}, "response": "95% CI ≈ [27.03, 32.71].", "meta": {"id": "WUA_STAT_25K_012824", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.28, s=14.0, n=16, compute CI."}]}, "response": "95% CI ≈ [16.42, 30.14].", "meta": {"id": "WUA_STAT_25K_012825", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.69, compute E and Var."}]}, "response": "E[X]=11.04; Var[X]=3.422.", "meta": {"id": "WUA_STAT_25K_012826", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.57, compute E and Var."}]}, "response": "E[X]=9.69; Var[X]=4.167.", "meta": {"id": "WUA_STAT_25K_012827", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.8, s=38.78, n=179, compute CI."}]}, "response": "95% CI ≈ [56.12, 67.48].", "meta": {"id": "WUA_STAT_25K_012828", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=47.42, s=22.79, n=189, compute CI."}]}, "response": "95% CI ≈ [44.17, 50.67].", "meta": {"id": "WUA_STAT_25K_012829", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=94.66, s=1.06, n=180, compute CI."}]}, "response": "95% CI ≈ [94.51, 94.81].", "meta": {"id": "WUA_STAT_25K_012830", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.38, s=16.23, n=158, compute CI."}]}, "response": "95% CI ≈ [160.85, 165.91].", "meta": {"id": "WUA_STAT_25K_012831", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.1, s=7.58, n=92, compute CI."}]}, "response": "95% CI ≈ [75.55, 78.65].", "meta": {"id": "WUA_STAT_25K_012832", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.85, s=10.85, n=113, compute CI."}]}, "response": "95% CI ≈ [161.85, 165.85].", "meta": {"id": "WUA_STAT_25K_012833", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.09, μ=67.49, σ=11.16, compute z."}]}, "response": "z = -4.427.", "meta": {"id": "WUA_STAT_25K_012834", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.5, compute E and Var."}]}, "response": "E[X]=4.00; Var[X]=2.000.", "meta": {"id": "WUA_STAT_25K_012835", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.72, s=9.91, n=149, compute CI."}]}, "response": "95% CI ≈ [86.13, 89.31].", "meta": {"id": "WUA_STAT_25K_012836", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.94, μ=25.83, σ=1.81, compute z."}]}, "response": "z = 26.580.", "meta": {"id": "WUA_STAT_25K_012837", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.184, sens=0.82, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.673.", "meta": {"id": "WUA_STAT_25K_012838", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.98, s=36.79, n=179, compute CI."}]}, "response": "95% CI ≈ [147.59, 158.37].", "meta": {"id": "WUA_STAT_25K_012839", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.84, s=17.25, n=113, compute CI."}]}, "response": "95% CI ≈ [22.66, 29.02].", "meta": {"id": "WUA_STAT_25K_012840", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=128.12, s=12.28, n=194, compute CI."}]}, "response": "95% CI ≈ [126.39, 129.85].", "meta": {"id": "WUA_STAT_25K_012841", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.84, μ=55.05, σ=8.96, compute z."}]}, "response": "z = 2.432.", "meta": {"id": "WUA_STAT_25K_012842", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.76, compute E and Var."}]}, "response": "E[X]=22.04; Var[X]=5.290.", "meta": {"id": "WUA_STAT_25K_012843", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.63, compute E and Var."}]}, "response": "E[X]=15.12; Var[X]=5.594.", "meta": {"id": "WUA_STAT_25K_012844", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.33, s=36.36, n=35, compute CI."}]}, "response": "95% CI ≈ [99.28, 123.38].", "meta": {"id": "WUA_STAT_25K_012845", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.46, compute E and Var."}]}, "response": "E[X]=10.58; Var[X]=5.713.", "meta": {"id": "WUA_STAT_25K_012846", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.31, compute E and Var."}]}, "response": "E[X]=2.48; Var[X]=1.711.", "meta": {"id": "WUA_STAT_25K_012847", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.79, compute E and Var."}]}, "response": "E[X]=22.12; Var[X]=4.645.", "meta": {"id": "WUA_STAT_25K_012848", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.98, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.445.", "meta": {"id": "WUA_STAT_25K_012849", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.75, s=19.63, n=122, compute CI."}]}, "response": "95% CI ≈ [71.27, 78.23].", "meta": {"id": "WUA_STAT_25K_012850", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.043, sens=0.97, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.187.", "meta": {"id": "WUA_STAT_25K_012851", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.06, sens=0.77, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.381.", "meta": {"id": "WUA_STAT_25K_012852", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.184, sens=0.97, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.510.", "meta": {"id": "WUA_STAT_25K_012853", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.85, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.492.", "meta": {"id": "WUA_STAT_25K_012854", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.65, s=19.76, n=136, compute CI."}]}, "response": "95% CI ≈ [73.33, 79.97].", "meta": {"id": "WUA_STAT_25K_012855", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.022, sens=0.71, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.058.", "meta": {"id": "WUA_STAT_25K_012856", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.84, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.576.", "meta": {"id": "WUA_STAT_25K_012857", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.088, sens=0.84, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.367.", "meta": {"id": "WUA_STAT_25K_012858", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.54, μ=16.45, σ=2.39, compute z."}]}, "response": "z = 2.130.", "meta": {"id": "WUA_STAT_25K_012859", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.46, μ=3.03, σ=8.04, compute z."}]}, "response": "z = -0.195.", "meta": {"id": "WUA_STAT_25K_012860", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.106, sens=0.73, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.230.", "meta": {"id": "WUA_STAT_25K_012861", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.47, compute E and Var."}]}, "response": "E[X]=4.70; Var[X]=2.491.", "meta": {"id": "WUA_STAT_25K_012862", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.71, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.844.", "meta": {"id": "WUA_STAT_25K_012863", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.32, μ=2.71, σ=8.44, compute z."}]}, "response": "z = 1.968.", "meta": {"id": "WUA_STAT_25K_012864", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.97, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.234.", "meta": {"id": "WUA_STAT_25K_012865", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.85, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.352.", "meta": {"id": "WUA_STAT_25K_012866", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.24, s=8.16, n=157, compute CI."}]}, "response": "95% CI ≈ [120.96, 123.52].", "meta": {"id": "WUA_STAT_25K_012867", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=26.85, s=18.57, n=57, compute CI."}]}, "response": "95% CI ≈ [22.03, 31.67].", "meta": {"id": "WUA_STAT_25K_012868", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.21, sens=0.7, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.674.", "meta": {"id": "WUA_STAT_25K_012869", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.042, sens=0.73, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.444.", "meta": {"id": "WUA_STAT_25K_012870", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.63, μ=9.76, σ=18.93, compute z."}]}, "response": "z = 2.740.", "meta": {"id": "WUA_STAT_25K_012871", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.93, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.540.", "meta": {"id": "WUA_STAT_25K_012872", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.25, s=25.88, n=111, compute CI."}]}, "response": "95% CI ≈ [80.44, 90.06].", "meta": {"id": "WUA_STAT_25K_012873", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.8, s=7.94, n=188, compute CI."}]}, "response": "95% CI ≈ [143.66, 145.94].", "meta": {"id": "WUA_STAT_25K_012874", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.49, compute E and Var."}]}, "response": "E[X]=2.94; Var[X]=1.499.", "meta": {"id": "WUA_STAT_25K_012875", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.16, compute E and Var."}]}, "response": "E[X]=4.64; Var[X]=3.898.", "meta": {"id": "WUA_STAT_25K_012876", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.33, compute E and Var."}]}, "response": "E[X]=7.92; Var[X]=5.306.", "meta": {"id": "WUA_STAT_25K_012877", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.46, s=19.14, n=197, compute CI."}]}, "response": "95% CI ≈ [40.79, 46.13].", "meta": {"id": "WUA_STAT_25K_012878", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.11, μ=2.1, σ=20.35, compute z."}]}, "response": "z = 3.195.", "meta": {"id": "WUA_STAT_25K_012879", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.76, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.174.", "meta": {"id": "WUA_STAT_25K_012880", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.17, μ=22.96, σ=3.61, compute z."}]}, "response": "z = -1.327.", "meta": {"id": "WUA_STAT_25K_012881", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.06, sens=0.96, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.290.", "meta": {"id": "WUA_STAT_25K_012882", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.24, μ=51.45, σ=21.86, compute z."}]}, "response": "z = -0.330.", "meta": {"id": "WUA_STAT_25K_012883", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.54, compute E and Var."}]}, "response": "E[X]=3.78; Var[X]=1.739.", "meta": {"id": "WUA_STAT_25K_012884", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.63, compute E and Var."}]}, "response": "E[X]=6.30; Var[X]=2.331.", "meta": {"id": "WUA_STAT_25K_012885", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.29, μ=3.46, σ=8.99, compute z."}]}, "response": "z = 1.872.", "meta": {"id": "WUA_STAT_25K_012886", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.93, μ=17.25, σ=22.01, compute z."}]}, "response": "z = 2.530.", "meta": {"id": "WUA_STAT_25K_012887", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.23, compute E and Var."}]}, "response": "E[X]=8.97; Var[X]=6.907.", "meta": {"id": "WUA_STAT_25K_012888", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.55, μ=40.8, σ=10.89, compute z."}]}, "response": "z = 3.650.", "meta": {"id": "WUA_STAT_25K_012889", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=93.86, s=29.18, n=91, compute CI."}]}, "response": "95% CI ≈ [87.86, 99.86].", "meta": {"id": "WUA_STAT_25K_012890", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.6, s=17.88, n=57, compute CI."}]}, "response": "95% CI ≈ [16.96, 26.24].", "meta": {"id": "WUA_STAT_25K_012891", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.59, compute E and Var."}]}, "response": "E[X]=16.52; Var[X]=6.773.", "meta": {"id": "WUA_STAT_25K_012892", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.89, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.199.", "meta": {"id": "WUA_STAT_25K_012893", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.71, s=9.91, n=91, compute CI."}]}, "response": "95% CI ≈ [140.67, 144.75].", "meta": {"id": "WUA_STAT_25K_012894", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.6, μ=11.98, σ=3.43, compute z."}]}, "response": "z = 11.843.", "meta": {"id": "WUA_STAT_25K_012895", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.99, s=13.79, n=83, compute CI."}]}, "response": "95% CI ≈ [43.02, 48.96].", "meta": {"id": "WUA_STAT_25K_012896", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.09, sens=0.84, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.509.", "meta": {"id": "WUA_STAT_25K_012897", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.3, μ=27.46, σ=2.2, compute z."}]}, "response": "z = 4.018.", "meta": {"id": "WUA_STAT_25K_012898", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.61, compute E and Var."}]}, "response": "E[X]=20.74; Var[X]=8.089.", "meta": {"id": "WUA_STAT_25K_012899", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.46, s=1.15, n=46, compute CI."}]}, "response": "95% CI ≈ [177.13, 177.79].", "meta": {"id": "WUA_STAT_25K_012900", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.51, μ=31.01, σ=3.85, compute z."}]}, "response": "z = 14.675.", "meta": {"id": "WUA_STAT_25K_012901", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.31, s=32.29, n=152, compute CI."}]}, "response": "95% CI ≈ [84.18, 94.44].", "meta": {"id": "WUA_STAT_25K_012902", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.1, s=9.02, n=28, compute CI."}]}, "response": "95% CI ≈ [81.76, 88.44].", "meta": {"id": "WUA_STAT_25K_012903", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.35, compute E and Var."}]}, "response": "E[X]=5.95; Var[X]=3.867.", "meta": {"id": "WUA_STAT_25K_012904", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.028, sens=0.98, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.320.", "meta": {"id": "WUA_STAT_25K_012905", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.39, μ=77.73, σ=2.71, compute z."}]}, "response": "z = -11.565.", "meta": {"id": "WUA_STAT_25K_012906", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.1, compute E and Var."}]}, "response": "E[X]=3.80; Var[X]=3.420.", "meta": {"id": "WUA_STAT_25K_012907", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.59, compute E and Var."}]}, "response": "E[X]=20.06; Var[X]=8.225.", "meta": {"id": "WUA_STAT_25K_012908", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=32.63, s=8.65, n=26, compute CI."}]}, "response": "95% CI ≈ [29.31, 35.95].", "meta": {"id": "WUA_STAT_25K_012909", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.25, μ=55.28, σ=9.08, compute z."}]}, "response": "z = -4.188.", "meta": {"id": "WUA_STAT_25K_012910", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.43, compute E and Var."}]}, "response": "E[X]=12.47; Var[X]=7.108.", "meta": {"id": "WUA_STAT_25K_012911", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.77, compute E and Var."}]}, "response": "E[X]=13.09; Var[X]=3.011.", "meta": {"id": "WUA_STAT_25K_012912", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.39, compute E and Var."}]}, "response": "E[X]=3.12; Var[X]=1.903.", "meta": {"id": "WUA_STAT_25K_012913", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.11, μ=83.41, σ=18.79, compute z."}]}, "response": "z = -3.954.", "meta": {"id": "WUA_STAT_25K_012914", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.1, μ=26.25, σ=1.13, compute z."}]}, "response": "z = -4.558.", "meta": {"id": "WUA_STAT_25K_012915", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.68, s=3.2, n=106, compute CI."}]}, "response": "95% CI ≈ [14.07, 15.29].", "meta": {"id": "WUA_STAT_25K_012916", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.83, μ=27.93, σ=21.46, compute z."}]}, "response": "z = 0.461.", "meta": {"id": "WUA_STAT_25K_012917", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.29, μ=13.38, σ=1.36, compute z."}]}, "response": "z = 23.463.", "meta": {"id": "WUA_STAT_25K_012918", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=169.97, s=24.89, n=39, compute CI."}]}, "response": "95% CI ≈ [162.16, 177.78].", "meta": {"id": "WUA_STAT_25K_012919", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.46, compute E and Var."}]}, "response": "E[X]=2.76; Var[X]=1.490.", "meta": {"id": "WUA_STAT_25K_012920", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.01, μ=81.61, σ=12.62, compute z."}]}, "response": "z = -1.712.", "meta": {"id": "WUA_STAT_25K_012921", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=126.67, s=4.84, n=149, compute CI."}]}, "response": "95% CI ≈ [125.89, 127.45].", "meta": {"id": "WUA_STAT_25K_012922", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.1, compute E and Var."}]}, "response": "E[X]=0.90; Var[X]=0.810.", "meta": {"id": "WUA_STAT_25K_012923", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.1, s=11.11, n=64, compute CI."}]}, "response": "95% CI ≈ [181.38, 186.82].", "meta": {"id": "WUA_STAT_25K_012924", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.17, s=29.07, n=182, compute CI."}]}, "response": "95% CI ≈ [105.95, 114.39].", "meta": {"id": "WUA_STAT_25K_012925", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.25, s=39.53, n=159, compute CI."}]}, "response": "95% CI ≈ [188.11, 200.39].", "meta": {"id": "WUA_STAT_25K_012926", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.75, s=19.71, n=16, compute CI."}]}, "response": "95% CI ≈ [178.09, 197.41].", "meta": {"id": "WUA_STAT_25K_012927", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.77, compute E and Var."}]}, "response": "E[X]=3.85; Var[X]=0.885.", "meta": {"id": "WUA_STAT_25K_012928", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.157, sens=0.95, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.558.", "meta": {"id": "WUA_STAT_25K_012929", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.44, s=13.62, n=64, compute CI."}]}, "response": "95% CI ≈ [118.10, 124.78].", "meta": {"id": "WUA_STAT_25K_012930", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.79, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.669.", "meta": {"id": "WUA_STAT_25K_012931", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.8, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.173.", "meta": {"id": "WUA_STAT_25K_012932", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.13, μ=60.97, σ=19.04, compute z."}]}, "response": "z = -1.987.", "meta": {"id": "WUA_STAT_25K_012933", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.211, sens=0.92, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.831.", "meta": {"id": "WUA_STAT_25K_012934", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.72, μ=64.58, σ=9.6, compute z."}]}, "response": "z = -0.194.", "meta": {"id": "WUA_STAT_25K_012935", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.62, compute E and Var."}]}, "response": "E[X]=21.08; Var[X]=8.010.", "meta": {"id": "WUA_STAT_25K_012936", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.041, sens=0.98, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.218.", "meta": {"id": "WUA_STAT_25K_012937", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.168, sens=0.89, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.375.", "meta": {"id": "WUA_STAT_25K_012938", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.19, compute E and Var."}]}, "response": "E[X]=4.75; Var[X]=3.848.", "meta": {"id": "WUA_STAT_25K_012939", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.52, s=4.58, n=96, compute CI."}]}, "response": "95% CI ≈ [128.60, 130.44].", "meta": {"id": "WUA_STAT_25K_012940", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.31, compute E and Var."}]}, "response": "E[X]=4.65; Var[X]=3.208.", "meta": {"id": "WUA_STAT_25K_012941", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.89, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.885.", "meta": {"id": "WUA_STAT_25K_012942", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.17, μ=63.83, σ=18.65, compute z."}]}, "response": "z = -1.590.", "meta": {"id": "WUA_STAT_25K_012943", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.5, compute E and Var."}]}, "response": "E[X]=11.50; Var[X]=5.750.", "meta": {"id": "WUA_STAT_25K_012944", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.58, μ=61.07, σ=19.95, compute z."}]}, "response": "z = 1.229.", "meta": {"id": "WUA_STAT_25K_012945", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.87, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.110.", "meta": {"id": "WUA_STAT_25K_012946", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.225, sens=0.78, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.507.", "meta": {"id": "WUA_STAT_25K_012947", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.61, compute E and Var."}]}, "response": "E[X]=13.42; Var[X]=5.234.", "meta": {"id": "WUA_STAT_25K_012948", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.133, sens=0.85, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.372.", "meta": {"id": "WUA_STAT_25K_012949", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.75, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.246.", "meta": {"id": "WUA_STAT_25K_012950", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=28.67, s=17.29, n=63, compute CI."}]}, "response": "95% CI ≈ [24.40, 32.94].", "meta": {"id": "WUA_STAT_25K_012951", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.96, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.329.", "meta": {"id": "WUA_STAT_25K_012952", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.01, s=4.46, n=57, compute CI."}]}, "response": "95% CI ≈ [83.85, 86.17].", "meta": {"id": "WUA_STAT_25K_012953", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.88, μ=83.89, σ=2.22, compute z."}]}, "response": "z = -20.725.", "meta": {"id": "WUA_STAT_25K_012954", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.46, μ=89.18, σ=14.23, compute z."}]}, "response": "z = 0.090.", "meta": {"id": "WUA_STAT_25K_012955", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.91, μ=52.75, σ=7.4, compute z."}]}, "response": "z = 3.130.", "meta": {"id": "WUA_STAT_25K_012956", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.24, sens=0.98, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.775.", "meta": {"id": "WUA_STAT_25K_012957", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.33, compute E and Var."}]}, "response": "E[X]=2.97; Var[X]=1.990.", "meta": {"id": "WUA_STAT_25K_012958", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.6, μ=1.6, σ=13.33, compute z."}]}, "response": "z = 5.776.", "meta": {"id": "WUA_STAT_25K_012959", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.45, s=11.06, n=115, compute CI."}]}, "response": "95% CI ≈ [134.43, 138.47].", "meta": {"id": "WUA_STAT_25K_012960", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.44, compute E and Var."}]}, "response": "E[X]=3.08; Var[X]=1.725.", "meta": {"id": "WUA_STAT_25K_012961", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.0, μ=99.41, σ=1.91, compute z."}]}, "response": "z = -24.822.", "meta": {"id": "WUA_STAT_25K_012962", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.33, s=20.29, n=13, compute CI."}]}, "response": "95% CI ≈ [143.30, 165.36].", "meta": {"id": "WUA_STAT_25K_012963", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.34, μ=90.28, σ=19.42, compute z."}]}, "response": "z = 0.364.", "meta": {"id": "WUA_STAT_25K_012964", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.76, compute E and Var."}]}, "response": "E[X]=26.60; Var[X]=6.384.", "meta": {"id": "WUA_STAT_25K_012965", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.75, compute E and Var."}]}, "response": "E[X]=18.00; Var[X]=4.500.", "meta": {"id": "WUA_STAT_25K_012966", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.86, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.272.", "meta": {"id": "WUA_STAT_25K_012967", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.85, μ=98.56, σ=14.17, compute z."}]}, "response": "z = -3.508.", "meta": {"id": "WUA_STAT_25K_012968", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.86, μ=56.86, σ=2.14, compute z."}]}, "response": "z = -9.813.", "meta": {"id": "WUA_STAT_25K_012969", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.17, sens=0.96, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.421.", "meta": {"id": "WUA_STAT_25K_012970", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.191, sens=0.88, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.675.", "meta": {"id": "WUA_STAT_25K_012971", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.21, μ=27.92, σ=9.8, compute z."}]}, "response": "z = 6.458.", "meta": {"id": "WUA_STAT_25K_012972", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.97, s=30.14, n=99, compute CI."}]}, "response": "95% CI ≈ [108.03, 119.91].", "meta": {"id": "WUA_STAT_25K_012973", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.39, μ=84.99, σ=7.47, compute z."}]}, "response": "z = -6.238.", "meta": {"id": "WUA_STAT_25K_012974", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=115.48, s=10.89, n=177, compute CI."}]}, "response": "95% CI ≈ [113.88, 117.08].", "meta": {"id": "WUA_STAT_25K_012975", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.44, compute E and Var."}]}, "response": "E[X]=15.84; Var[X]=8.870.", "meta": {"id": "WUA_STAT_25K_012976", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.31, μ=70.49, σ=20.8, compute z."}]}, "response": "z = -1.114.", "meta": {"id": "WUA_STAT_25K_012977", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.2, compute E and Var."}]}, "response": "E[X]=2.20; Var[X]=1.760.", "meta": {"id": "WUA_STAT_25K_012978", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.04, s=21.4, n=144, compute CI."}]}, "response": "95% CI ≈ [181.54, 188.54].", "meta": {"id": "WUA_STAT_25K_012979", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.46, compute E and Var."}]}, "response": "E[X]=4.60; Var[X]=2.484.", "meta": {"id": "WUA_STAT_25K_012980", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.89, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.537.", "meta": {"id": "WUA_STAT_25K_012981", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.24, compute E and Var."}]}, "response": "E[X]=5.04; Var[X]=3.830.", "meta": {"id": "WUA_STAT_25K_012982", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.0, s=15.0, n=108, compute CI."}]}, "response": "95% CI ≈ [74.17, 79.83].", "meta": {"id": "WUA_STAT_25K_012983", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.248, sens=0.78, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.837.", "meta": {"id": "WUA_STAT_25K_012984", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.3, μ=23.07, σ=16.34, compute z."}]}, "response": "z = 2.829.", "meta": {"id": "WUA_STAT_25K_012985", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.11, compute E and Var."}]}, "response": "E[X]=3.96; Var[X]=3.524.", "meta": {"id": "WUA_STAT_25K_012986", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.221, sens=0.97, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.514.", "meta": {"id": "WUA_STAT_25K_012987", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.06, s=39.49, n=113, compute CI."}]}, "response": "95% CI ≈ [112.78, 127.34].", "meta": {"id": "WUA_STAT_25K_012988", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.5, s=9.99, n=48, compute CI."}]}, "response": "95% CI ≈ [62.67, 68.33].", "meta": {"id": "WUA_STAT_25K_012989", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.46, s=21.33, n=104, compute CI."}]}, "response": "95% CI ≈ [119.36, 127.56].", "meta": {"id": "WUA_STAT_25K_012990", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.02, s=5.36, n=70, compute CI."}]}, "response": "95% CI ≈ [178.76, 181.28].", "meta": {"id": "WUA_STAT_25K_012991", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.57, s=14.28, n=102, compute CI."}]}, "response": "95% CI ≈ [20.80, 26.34].", "meta": {"id": "WUA_STAT_25K_012992", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.02, μ=67.47, σ=9.91, compute z."}]}, "response": "z = 1.771.", "meta": {"id": "WUA_STAT_25K_012993", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.07, s=21.27, n=82, compute CI."}]}, "response": "95% CI ≈ [24.47, 33.67].", "meta": {"id": "WUA_STAT_25K_012994", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.42, μ=60.85, σ=23.71, compute z."}]}, "response": "z = -0.862.", "meta": {"id": "WUA_STAT_25K_012995", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.72, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.576.", "meta": {"id": "WUA_STAT_25K_012996", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.191, sens=0.96, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.486.", "meta": {"id": "WUA_STAT_25K_012997", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.68, compute E and Var."}]}, "response": "E[X]=3.40; Var[X]=1.088.", "meta": {"id": "WUA_STAT_25K_012998", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=73.8, s=38.38, n=20, compute CI."}]}, "response": "95% CI ≈ [56.98, 90.62].", "meta": {"id": "WUA_STAT_25K_012999", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.29, compute E and Var."}]}, "response": "E[X]=4.93; Var[X]=3.500.", "meta": {"id": "WUA_STAT_25K_013000", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.81, μ=33.79, σ=12.73, compute z."}]}, "response": "z = 4.401.", "meta": {"id": "WUA_STAT_25K_013001", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.95, μ=83.55, σ=9.81, compute z."}]}, "response": "z = -1.488.", "meta": {"id": "WUA_STAT_25K_013002", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.14, compute E and Var."}]}, "response": "E[X]=5.32; Var[X]=4.575.", "meta": {"id": "WUA_STAT_25K_013003", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.27, μ=30.12, σ=10.8, compute z."}]}, "response": "z = -0.079.", "meta": {"id": "WUA_STAT_25K_013004", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.73, compute E and Var."}]}, "response": "E[X]=9.49; Var[X]=2.562.", "meta": {"id": "WUA_STAT_25K_013005", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.019, sens=0.78, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.059.", "meta": {"id": "WUA_STAT_25K_013006", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.12, μ=74.22, σ=24.11, compute z."}]}, "response": "z = 0.452.", "meta": {"id": "WUA_STAT_25K_013007", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.87, μ=15.89, σ=17.16, compute z."}]}, "response": "z = 2.330.", "meta": {"id": "WUA_STAT_25K_013008", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=183.44, s=30.55, n=179, compute CI."}]}, "response": "95% CI ≈ [178.96, 187.92].", "meta": {"id": "WUA_STAT_25K_013009", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=183.6, s=30.5, n=125, compute CI."}]}, "response": "95% CI ≈ [178.25, 188.95].", "meta": {"id": "WUA_STAT_25K_013010", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=126.86, s=9.91, n=149, compute CI."}]}, "response": "95% CI ≈ [125.27, 128.45].", "meta": {"id": "WUA_STAT_25K_013011", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.99, μ=65.29, σ=23.77, compute z."}]}, "response": "z = -1.780.", "meta": {"id": "WUA_STAT_25K_013012", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.62, compute E and Var."}]}, "response": "E[X]=23.56; Var[X]=8.953.", "meta": {"id": "WUA_STAT_25K_013013", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.72, μ=4.03, σ=22.32, compute z."}]}, "response": "z = 3.212.", "meta": {"id": "WUA_STAT_25K_013014", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.93, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.068.", "meta": {"id": "WUA_STAT_25K_013015", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.28, μ=34.67, σ=22.08, compute z."}]}, "response": "z = -0.425.", "meta": {"id": "WUA_STAT_25K_013016", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.76, s=20.07, n=166, compute CI."}]}, "response": "95% CI ≈ [28.71, 34.81].", "meta": {"id": "WUA_STAT_25K_013017", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.65, compute E and Var."}]}, "response": "E[X]=12.35; Var[X]=4.322.", "meta": {"id": "WUA_STAT_25K_013018", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.229, sens=0.95, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.758.", "meta": {"id": "WUA_STAT_25K_013019", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.84, μ=60.45, σ=21.36, compute z."}]}, "response": "z = -1.854.", "meta": {"id": "WUA_STAT_25K_013020", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.93, μ=59.37, σ=1.79, compute z."}]}, "response": "z = 0.872.", "meta": {"id": "WUA_STAT_25K_013021", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.79, compute E and Var."}]}, "response": "E[X]=12.64; Var[X]=2.654.", "meta": {"id": "WUA_STAT_25K_013022", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.173, sens=0.88, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.467.", "meta": {"id": "WUA_STAT_25K_013023", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.165, sens=0.76, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.938.", "meta": {"id": "WUA_STAT_25K_013024", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.71, μ=13.73, σ=23.92, compute z."}]}, "response": "z = -0.544.", "meta": {"id": "WUA_STAT_25K_013025", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.76, compute E and Var."}]}, "response": "E[X]=26.60; Var[X]=6.384.", "meta": {"id": "WUA_STAT_25K_013026", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.229, sens=0.88, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.744.", "meta": {"id": "WUA_STAT_25K_013027", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.3, μ=99.61, σ=6.44, compute z."}]}, "response": "z = -14.955.", "meta": {"id": "WUA_STAT_25K_013028", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.08, s=39.45, n=122, compute CI."}]}, "response": "95% CI ≈ [170.08, 184.08].", "meta": {"id": "WUA_STAT_25K_013029", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.1, s=11.44, n=46, compute CI."}]}, "response": "95% CI ≈ [7.79, 14.41].", "meta": {"id": "WUA_STAT_25K_013030", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.86, μ=2.87, σ=19.51, compute z."}]}, "response": "z = 4.869.", "meta": {"id": "WUA_STAT_25K_013031", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.63, s=26.35, n=90, compute CI."}]}, "response": "95% CI ≈ [161.19, 172.07].", "meta": {"id": "WUA_STAT_25K_013032", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.08, compute E and Var."}]}, "response": "E[X]=2.88; Var[X]=2.650.", "meta": {"id": "WUA_STAT_25K_013033", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.75, compute E and Var."}]}, "response": "E[X]=18.75; Var[X]=4.688.", "meta": {"id": "WUA_STAT_25K_013034", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.03, μ=44.85, σ=3.74, compute z."}]}, "response": "z = -10.914.", "meta": {"id": "WUA_STAT_25K_013035", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.247, sens=0.95, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.862.", "meta": {"id": "WUA_STAT_25K_013036", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.25, s=11.7, n=125, compute CI."}]}, "response": "95% CI ≈ [110.20, 114.30].", "meta": {"id": "WUA_STAT_25K_013037", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.68, s=18.78, n=27, compute CI."}]}, "response": "95% CI ≈ [91.60, 105.76].", "meta": {"id": "WUA_STAT_25K_013038", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.032, sens=0.9, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.129.", "meta": {"id": "WUA_STAT_25K_013039", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.31, compute E and Var."}]}, "response": "E[X]=3.41; Var[X]=2.353.", "meta": {"id": "WUA_STAT_25K_013040", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.48, s=3.65, n=28, compute CI."}]}, "response": "95% CI ≈ [57.13, 59.83].", "meta": {"id": "WUA_STAT_25K_013041", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.38, compute E and Var."}]}, "response": "E[X]=11.40; Var[X]=7.068.", "meta": {"id": "WUA_STAT_25K_013042", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.47, compute E and Var."}]}, "response": "E[X]=2.82; Var[X]=1.495.", "meta": {"id": "WUA_STAT_25K_013043", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.13, compute E and Var."}]}, "response": "E[X]=2.99; Var[X]=2.601.", "meta": {"id": "WUA_STAT_25K_013044", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.4, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_013045", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=20.62, s=15.89, n=181, compute CI."}]}, "response": "95% CI ≈ [18.31, 22.93].", "meta": {"id": "WUA_STAT_25K_013046", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.03, μ=58.42, σ=2.22, compute z."}]}, "response": "z = 4.779.", "meta": {"id": "WUA_STAT_25K_013047", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.58, s=32.97, n=19, compute CI."}]}, "response": "95% CI ≈ [18.75, 48.41].", "meta": {"id": "WUA_STAT_25K_013048", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.066, sens=0.9, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.366.", "meta": {"id": "WUA_STAT_25K_013049", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.28, μ=91.78, σ=17.6, compute z."}]}, "response": "z = -4.062.", "meta": {"id": "WUA_STAT_25K_013050", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.15, compute E and Var."}]}, "response": "E[X]=5.10; Var[X]=4.335.", "meta": {"id": "WUA_STAT_25K_013051", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.46, compute E and Var."}]}, "response": "E[X]=5.06; Var[X]=2.732.", "meta": {"id": "WUA_STAT_25K_013052", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.01, μ=6.03, σ=8.93, compute z."}]}, "response": "z = 9.404.", "meta": {"id": "WUA_STAT_25K_013053", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.69, s=24.67, n=192, compute CI."}]}, "response": "95% CI ≈ [74.20, 81.18].", "meta": {"id": "WUA_STAT_25K_013054", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.97, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.546.", "meta": {"id": "WUA_STAT_25K_013055", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.88, μ=88.91, σ=11.62, compute z."}]}, "response": "z = -0.863.", "meta": {"id": "WUA_STAT_25K_013056", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.87, s=39.44, n=41, compute CI."}]}, "response": "95% CI ≈ [76.80, 100.94].", "meta": {"id": "WUA_STAT_25K_013057", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.021, sens=0.92, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.073.", "meta": {"id": "WUA_STAT_25K_013058", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.68, s=36.19, n=72, compute CI."}]}, "response": "95% CI ≈ [117.32, 134.04].", "meta": {"id": "WUA_STAT_25K_013059", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.58, compute E and Var."}]}, "response": "E[X]=16.82; Var[X]=7.064.", "meta": {"id": "WUA_STAT_25K_013060", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.02, s=19.37, n=42, compute CI."}]}, "response": "95% CI ≈ [167.16, 178.88].", "meta": {"id": "WUA_STAT_25K_013061", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.02, μ=74.43, σ=18.85, compute z."}]}, "response": "z = -3.841.", "meta": {"id": "WUA_STAT_25K_013062", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.5, μ=27.46, σ=2.45, compute z."}]}, "response": "z = -5.698.", "meta": {"id": "WUA_STAT_25K_013063", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.65, s=31.0, n=132, compute CI."}]}, "response": "95% CI ≈ [7.36, 17.94].", "meta": {"id": "WUA_STAT_25K_013064", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.61, μ=98.35, σ=4.01, compute z."}]}, "response": "z = -23.377.", "meta": {"id": "WUA_STAT_25K_013065", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.67, s=26.94, n=49, compute CI."}]}, "response": "95% CI ≈ [166.13, 181.21].", "meta": {"id": "WUA_STAT_25K_013066", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.105, sens=0.8, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.439.", "meta": {"id": "WUA_STAT_25K_013067", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.16, μ=1.17, σ=19.3, compute z."}]}, "response": "z = 4.455.", "meta": {"id": "WUA_STAT_25K_013068", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.81, μ=11.37, σ=2.17, compute z."}]}, "response": "z = 32.000.", "meta": {"id": "WUA_STAT_25K_013069", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.018, sens=0.91, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.063.", "meta": {"id": "WUA_STAT_25K_013070", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.5, compute E and Var."}]}, "response": "E[X]=13.50; Var[X]=6.750.", "meta": {"id": "WUA_STAT_25K_013071", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.2, compute E and Var."}]}, "response": "E[X]=5.40; Var[X]=4.320.", "meta": {"id": "WUA_STAT_25K_013072", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.42, s=35.2, n=14, compute CI."}]}, "response": "95% CI ≈ [167.98, 204.86].", "meta": {"id": "WUA_STAT_25K_013073", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.149, sens=0.83, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.618.", "meta": {"id": "WUA_STAT_25K_013074", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.191, sens=0.72, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.405.", "meta": {"id": "WUA_STAT_25K_013075", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.16, compute E and Var."}]}, "response": "E[X]=3.20; Var[X]=2.688.", "meta": {"id": "WUA_STAT_25K_013076", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.29, s=30.62, n=176, compute CI."}]}, "response": "95% CI ≈ [170.77, 179.81].", "meta": {"id": "WUA_STAT_25K_013077", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.53, s=6.64, n=22, compute CI."}]}, "response": "95% CI ≈ [43.76, 49.30].", "meta": {"id": "WUA_STAT_25K_013078", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.6, μ=10.99, σ=11.01, compute z."}]}, "response": "z = 4.233.", "meta": {"id": "WUA_STAT_25K_013079", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.71, compute E and Var."}]}, "response": "E[X]=21.30; Var[X]=6.177.", "meta": {"id": "WUA_STAT_25K_013080", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.9, s=27.51, n=93, compute CI."}]}, "response": "95% CI ≈ [26.31, 37.49].", "meta": {"id": "WUA_STAT_25K_013081", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.34, compute E and Var."}]}, "response": "E[X]=7.48; Var[X]=4.937.", "meta": {"id": "WUA_STAT_25K_013082", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.15, μ=50.66, σ=15.65, compute z."}]}, "response": "z = -0.160.", "meta": {"id": "WUA_STAT_25K_013083", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.24, s=17.96, n=115, compute CI."}]}, "response": "95% CI ≈ [174.96, 181.52].", "meta": {"id": "WUA_STAT_25K_013084", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.03, μ=28.02, σ=6.64, compute z."}]}, "response": "z = 3.315.", "meta": {"id": "WUA_STAT_25K_013085", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.12, sens=0.77, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.344.", "meta": {"id": "WUA_STAT_25K_013086", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.195, sens=0.72, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.392.", "meta": {"id": "WUA_STAT_25K_013087", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.89, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.836.", "meta": {"id": "WUA_STAT_25K_013088", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.65, μ=72.06, σ=21.93, compute z."}]}, "response": "z = -0.566.", "meta": {"id": "WUA_STAT_25K_013089", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.69, μ=66.59, σ=7.51, compute z."}]}, "response": "z = -1.718.", "meta": {"id": "WUA_STAT_25K_013090", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.66, s=12.34, n=10, compute CI."}]}, "response": "95% CI ≈ [163.01, 178.31].", "meta": {"id": "WUA_STAT_25K_013091", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=95.0, s=31.64, n=80, compute CI."}]}, "response": "95% CI ≈ [88.07, 101.93].", "meta": {"id": "WUA_STAT_25K_013092", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.19, s=25.4, n=29, compute CI."}]}, "response": "95% CI ≈ [115.95, 134.43].", "meta": {"id": "WUA_STAT_25K_013093", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.6, compute E and Var."}]}, "response": "E[X]=13.20; Var[X]=5.280.", "meta": {"id": "WUA_STAT_25K_013094", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.88, s=31.53, n=88, compute CI."}]}, "response": "95% CI ≈ [172.29, 185.47].", "meta": {"id": "WUA_STAT_25K_013095", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.73, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.588.", "meta": {"id": "WUA_STAT_25K_013096", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.037, sens=0.76, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.094.", "meta": {"id": "WUA_STAT_25K_013097", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.9, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.337.", "meta": {"id": "WUA_STAT_25K_013098", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.29, s=19.03, n=47, compute CI."}]}, "response": "95% CI ≈ [174.85, 185.73].", "meta": {"id": "WUA_STAT_25K_013099", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.8, s=37.1, n=41, compute CI."}]}, "response": "95% CI ≈ [95.44, 118.16].", "meta": {"id": "WUA_STAT_25K_013100", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.197, sens=0.77, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.451.", "meta": {"id": "WUA_STAT_25K_013101", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.33, s=28.05, n=87, compute CI."}]}, "response": "95% CI ≈ [78.44, 90.22].", "meta": {"id": "WUA_STAT_25K_013102", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.26, μ=33.06, σ=10.99, compute z."}]}, "response": "z = -2.530.", "meta": {"id": "WUA_STAT_25K_013103", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.78, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.455.", "meta": {"id": "WUA_STAT_25K_013104", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.8, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.590.", "meta": {"id": "WUA_STAT_25K_013105", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.66, compute E and Var."}]}, "response": "E[X]=22.44; Var[X]=7.630.", "meta": {"id": "WUA_STAT_25K_013106", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.59, s=9.66, n=54, compute CI."}]}, "response": "95% CI ≈ [121.01, 126.17].", "meta": {"id": "WUA_STAT_25K_013107", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.16, μ=14.99, σ=20.24, compute z."}]}, "response": "z = 3.170.", "meta": {"id": "WUA_STAT_25K_013108", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.03, μ=64.25, σ=19.52, compute z."}]}, "response": "z = -1.958.", "meta": {"id": "WUA_STAT_25K_013109", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.1, μ=93.95, σ=3.22, compute z."}]}, "response": "z = -12.686.", "meta": {"id": "WUA_STAT_25K_013110", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.32, μ=77.15, σ=16.55, compute z."}]}, "response": "z = -3.676.", "meta": {"id": "WUA_STAT_25K_013111", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.77, compute E and Var."}]}, "response": "E[X]=3.85; Var[X]=0.885.", "meta": {"id": "WUA_STAT_25K_013112", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.7, compute E and Var."}]}, "response": "E[X]=4.20; Var[X]=1.260.", "meta": {"id": "WUA_STAT_25K_013113", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.33, compute E and Var."}]}, "response": "E[X]=2.97; Var[X]=1.990.", "meta": {"id": "WUA_STAT_25K_013114", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.61, s=19.26, n=43, compute CI."}]}, "response": "95% CI ≈ [126.85, 138.37].", "meta": {"id": "WUA_STAT_25K_013115", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.09, s=23.79, n=62, compute CI."}]}, "response": "95% CI ≈ [127.17, 139.01].", "meta": {"id": "WUA_STAT_25K_013116", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.85, s=17.24, n=142, compute CI."}]}, "response": "95% CI ≈ [164.01, 169.69].", "meta": {"id": "WUA_STAT_25K_013117", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.86, μ=32.68, σ=5.55, compute z."}]}, "response": "z = -4.652.", "meta": {"id": "WUA_STAT_25K_013118", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.87, s=1.61, n=70, compute CI."}]}, "response": "95% CI ≈ [63.49, 64.25].", "meta": {"id": "WUA_STAT_25K_013119", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.75, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.475.", "meta": {"id": "WUA_STAT_25K_013120", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.86, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.402.", "meta": {"id": "WUA_STAT_25K_013121", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.46, compute E and Var."}]}, "response": "E[X]=6.44; Var[X]=3.478.", "meta": {"id": "WUA_STAT_25K_013122", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.87, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.806.", "meta": {"id": "WUA_STAT_25K_013123", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.31, compute E and Var."}]}, "response": "E[X]=1.86; Var[X]=1.283.", "meta": {"id": "WUA_STAT_25K_013124", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.77, compute E and Var."}]}, "response": "E[X]=11.55; Var[X]=2.656.", "meta": {"id": "WUA_STAT_25K_013125", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.63, μ=41.66, σ=8.49, compute z."}]}, "response": "z = 1.645.", "meta": {"id": "WUA_STAT_25K_013126", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.5, s=33.31, n=17, compute CI."}]}, "response": "95% CI ≈ [177.67, 209.33].", "meta": {"id": "WUA_STAT_25K_013127", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.44, μ=5.56, σ=3.94, compute z."}]}, "response": "z = 12.406.", "meta": {"id": "WUA_STAT_25K_013128", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.73, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.454.", "meta": {"id": "WUA_STAT_25K_013129", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.38, compute E and Var."}]}, "response": "E[X]=14.06; Var[X]=8.717.", "meta": {"id": "WUA_STAT_25K_013130", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.96, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.511.", "meta": {"id": "WUA_STAT_25K_013131", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.221, sens=0.9, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.663.", "meta": {"id": "WUA_STAT_25K_013132", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.84, μ=79.38, σ=16.88, compute z."}]}, "response": "z = 0.383.", "meta": {"id": "WUA_STAT_25K_013133", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.48, μ=78.97, σ=2.43, compute z."}]}, "response": "z = -31.066.", "meta": {"id": "WUA_STAT_25K_013134", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.224, sens=0.88, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.536.", "meta": {"id": "WUA_STAT_25K_013135", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.76, μ=32.03, σ=13.19, compute z."}]}, "response": "z = 0.662.", "meta": {"id": "WUA_STAT_25K_013136", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.74, s=5.98, n=115, compute CI."}]}, "response": "95% CI ≈ [69.65, 71.83].", "meta": {"id": "WUA_STAT_25K_013137", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.42, compute E and Var."}]}, "response": "E[X]=8.82; Var[X]=5.116.", "meta": {"id": "WUA_STAT_25K_013138", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.32, compute E and Var."}]}, "response": "E[X]=2.24; Var[X]=1.523.", "meta": {"id": "WUA_STAT_25K_013139", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.65, compute E and Var."}]}, "response": "E[X]=13.65; Var[X]=4.777.", "meta": {"id": "WUA_STAT_25K_013140", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=73.63, s=3.67, n=66, compute CI."}]}, "response": "95% CI ≈ [72.74, 74.52].", "meta": {"id": "WUA_STAT_25K_013141", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.43, μ=13.58, σ=18.7, compute z."}]}, "response": "z = 1.757.", "meta": {"id": "WUA_STAT_25K_013142", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.72, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.110.", "meta": {"id": "WUA_STAT_25K_013143", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.78, μ=52.49, σ=9.97, compute z."}]}, "response": "z = -0.172.", "meta": {"id": "WUA_STAT_25K_013144", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.05, compute E and Var."}]}, "response": "E[X]=1.05; Var[X]=0.997.", "meta": {"id": "WUA_STAT_25K_013145", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.03, sens=0.84, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.157.", "meta": {"id": "WUA_STAT_25K_013146", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.085, sens=0.83, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.312.", "meta": {"id": "WUA_STAT_25K_013147", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.84, μ=26.0, σ=14.97, compute z."}]}, "response": "z = -1.079.", "meta": {"id": "WUA_STAT_25K_013148", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.51, μ=1.34, σ=15.45, compute z."}]}, "response": "z = 2.794.", "meta": {"id": "WUA_STAT_25K_013149", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.34, s=8.52, n=106, compute CI."}]}, "response": "95% CI ≈ [27.72, 30.96].", "meta": {"id": "WUA_STAT_25K_013150", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.57, μ=0.05, σ=21.9, compute z."}]}, "response": "z = 1.439.", "meta": {"id": "WUA_STAT_25K_013151", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.76, μ=88.58, σ=21.52, compute z."}]}, "response": "z = -1.014.", "meta": {"id": "WUA_STAT_25K_013152", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.03, s=7.42, n=10, compute CI."}]}, "response": "95% CI ≈ [18.43, 27.63].", "meta": {"id": "WUA_STAT_25K_013153", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.71, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.424.", "meta": {"id": "WUA_STAT_25K_013154", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.09, s=2.35, n=84, compute CI."}]}, "response": "95% CI ≈ [100.59, 101.59].", "meta": {"id": "WUA_STAT_25K_013155", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.47, compute E and Var."}]}, "response": "E[X]=9.40; Var[X]=4.982.", "meta": {"id": "WUA_STAT_25K_013156", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.43, compute E and Var."}]}, "response": "E[X]=7.74; Var[X]=4.412.", "meta": {"id": "WUA_STAT_25K_013157", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.122, sens=0.88, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.485.", "meta": {"id": "WUA_STAT_25K_013158", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.02, s=2.15, n=65, compute CI."}]}, "response": "95% CI ≈ [129.50, 130.54].", "meta": {"id": "WUA_STAT_25K_013159", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.185, sens=0.82, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.756.", "meta": {"id": "WUA_STAT_25K_013160", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.01, μ=94.7, σ=19.69, compute z."}]}, "response": "z = -4.555.", "meta": {"id": "WUA_STAT_25K_013161", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.09, compute E and Var."}]}, "response": "E[X]=3.42; Var[X]=3.112.", "meta": {"id": "WUA_STAT_25K_013162", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.88, μ=47.42, σ=24.98, compute z."}]}, "response": "z = 1.460.", "meta": {"id": "WUA_STAT_25K_013163", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=164.17, s=8.35, n=154, compute CI."}]}, "response": "95% CI ≈ [162.85, 165.49].", "meta": {"id": "WUA_STAT_25K_013164", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.36, compute E and Var."}]}, "response": "E[X]=11.52; Var[X]=7.373.", "meta": {"id": "WUA_STAT_25K_013165", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.87, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.589.", "meta": {"id": "WUA_STAT_25K_013166", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.31, compute E and Var."}]}, "response": "E[X]=11.47; Var[X]=7.914.", "meta": {"id": "WUA_STAT_25K_013167", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.072, sens=0.89, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.277.", "meta": {"id": "WUA_STAT_25K_013168", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.83, μ=74.28, σ=1.82, compute z."}]}, "response": "z = 7.995.", "meta": {"id": "WUA_STAT_25K_013169", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.244, sens=0.83, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.728.", "meta": {"id": "WUA_STAT_25K_013170", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.62, compute E and Var."}]}, "response": "E[X]=3.10; Var[X]=1.178.", "meta": {"id": "WUA_STAT_25K_013171", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.01, s=20.8, n=158, compute CI."}]}, "response": "95% CI ≈ [94.77, 101.25].", "meta": {"id": "WUA_STAT_25K_013172", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.8, μ=94.14, σ=4.88, compute z."}]}, "response": "z = -18.307.", "meta": {"id": "WUA_STAT_25K_013173", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.32, compute E and Var."}]}, "response": "E[X]=9.92; Var[X]=6.746.", "meta": {"id": "WUA_STAT_25K_013174", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.72, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.294.", "meta": {"id": "WUA_STAT_25K_013175", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.98, μ=70.81, σ=6.8, compute z."}]}, "response": "z = -4.975.", "meta": {"id": "WUA_STAT_25K_013176", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.32, μ=85.21, σ=2.21, compute z."}]}, "response": "z = -1.308.", "meta": {"id": "WUA_STAT_25K_013177", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.3, s=36.08, n=125, compute CI."}]}, "response": "95% CI ≈ [54.97, 67.63].", "meta": {"id": "WUA_STAT_25K_013178", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.84, s=5.05, n=17, compute CI."}]}, "response": "95% CI ≈ [112.44, 117.24].", "meta": {"id": "WUA_STAT_25K_013179", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.92, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.468.", "meta": {"id": "WUA_STAT_25K_013180", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.32, μ=22.49, σ=24.02, compute z."}]}, "response": "z = 1.284.", "meta": {"id": "WUA_STAT_25K_013181", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.16, compute E and Var."}]}, "response": "E[X]=0.96; Var[X]=0.806.", "meta": {"id": "WUA_STAT_25K_013182", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.6, s=35.12, n=132, compute CI."}]}, "response": "95% CI ≈ [71.61, 83.59].", "meta": {"id": "WUA_STAT_25K_013183", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.153, sens=0.95, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.462.", "meta": {"id": "WUA_STAT_25K_013184", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.53, μ=77.52, σ=11.59, compute z."}]}, "response": "z = -3.105.", "meta": {"id": "WUA_STAT_25K_013185", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.15, μ=79.93, σ=4.38, compute z."}]}, "response": "z = 4.388.", "meta": {"id": "WUA_STAT_25K_013186", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.39, s=32.01, n=117, compute CI."}]}, "response": "95% CI ≈ [86.59, 98.19].", "meta": {"id": "WUA_STAT_25K_013187", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.011, sens=0.83, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.117.", "meta": {"id": "WUA_STAT_25K_013188", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.08, μ=37.6, σ=6.76, compute z."}]}, "response": "z = -1.704.", "meta": {"id": "WUA_STAT_25K_013189", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.216, sens=0.95, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.579.", "meta": {"id": "WUA_STAT_25K_013190", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.83, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.660.", "meta": {"id": "WUA_STAT_25K_013191", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=162.85, s=6.64, n=86, compute CI."}]}, "response": "95% CI ≈ [161.45, 164.25].", "meta": {"id": "WUA_STAT_25K_013192", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.62, compute E and Var."}]}, "response": "E[X]=19.84; Var[X]=7.539.", "meta": {"id": "WUA_STAT_25K_013193", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.88, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.162.", "meta": {"id": "WUA_STAT_25K_013194", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.17, compute E and Var."}]}, "response": "E[X]=2.72; Var[X]=2.258.", "meta": {"id": "WUA_STAT_25K_013195", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.87, s=10.13, n=117, compute CI."}]}, "response": "95% CI ≈ [66.03, 69.71].", "meta": {"id": "WUA_STAT_25K_013196", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.021, sens=0.84, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.087.", "meta": {"id": "WUA_STAT_25K_013197", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.25, μ=85.53, σ=20.82, compute z."}]}, "response": "z = -1.406.", "meta": {"id": "WUA_STAT_25K_013198", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=128.28, s=17.73, n=65, compute CI."}]}, "response": "95% CI ≈ [123.97, 132.59].", "meta": {"id": "WUA_STAT_25K_013199", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.95, s=12.34, n=98, compute CI."}]}, "response": "95% CI ≈ [132.51, 137.39].", "meta": {"id": "WUA_STAT_25K_013200", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.52, compute E and Var."}]}, "response": "E[X]=19.24; Var[X]=9.235.", "meta": {"id": "WUA_STAT_25K_013201", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.43, μ=26.26, σ=8.35, compute z."}]}, "response": "z = -3.093.", "meta": {"id": "WUA_STAT_25K_013202", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.91, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.079.", "meta": {"id": "WUA_STAT_25K_013203", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.62, s=2.98, n=32, compute CI."}]}, "response": "95% CI ≈ [20.59, 22.65].", "meta": {"id": "WUA_STAT_25K_013204", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.59, s=13.69, n=78, compute CI."}]}, "response": "95% CI ≈ [72.55, 78.63].", "meta": {"id": "WUA_STAT_25K_013205", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.58, compute E and Var."}]}, "response": "E[X]=17.98; Var[X]=7.552.", "meta": {"id": "WUA_STAT_25K_013206", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.43, compute E and Var."}]}, "response": "E[X]=16.77; Var[X]=9.559.", "meta": {"id": "WUA_STAT_25K_013207", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.21, μ=94.99, σ=22.81, compute z."}]}, "response": "z = -2.358.", "meta": {"id": "WUA_STAT_25K_013208", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.91, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.373.", "meta": {"id": "WUA_STAT_25K_013209", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.69, μ=64.98, σ=20.37, compute z."}]}, "response": "z = -0.702.", "meta": {"id": "WUA_STAT_25K_013210", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.228, sens=0.99, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.619.", "meta": {"id": "WUA_STAT_25K_013211", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.39, μ=56.51, σ=24.95, compute z."}]}, "response": "z = -1.488.", "meta": {"id": "WUA_STAT_25K_013212", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.9, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.420.", "meta": {"id": "WUA_STAT_25K_013213", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.091, sens=0.79, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.283.", "meta": {"id": "WUA_STAT_25K_013214", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.38, s=24.84, n=177, compute CI."}]}, "response": "95% CI ≈ [7.72, 15.04].", "meta": {"id": "WUA_STAT_25K_013215", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.99, μ=65.73, σ=7.48, compute z."}]}, "response": "z = 1.906.", "meta": {"id": "WUA_STAT_25K_013216", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.87, μ=32.3, σ=15.15, compute z."}]}, "response": "z = -1.018.", "meta": {"id": "WUA_STAT_25K_013217", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.095, sens=0.93, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.317.", "meta": {"id": "WUA_STAT_25K_013218", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.52, μ=20.25, σ=23.64, compute z."}]}, "response": "z = 2.000.", "meta": {"id": "WUA_STAT_25K_013219", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.42, compute E and Var."}]}, "response": "E[X]=7.98; Var[X]=4.628.", "meta": {"id": "WUA_STAT_25K_013220", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.089, sens=0.89, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.491.", "meta": {"id": "WUA_STAT_25K_013221", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.15, μ=44.02, σ=6.87, compute z."}]}, "response": "z = -4.493.", "meta": {"id": "WUA_STAT_25K_013222", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.203, sens=0.84, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.425.", "meta": {"id": "WUA_STAT_25K_013223", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.21, compute E and Var."}]}, "response": "E[X]=1.68; Var[X]=1.327.", "meta": {"id": "WUA_STAT_25K_013224", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.72, compute E and Var."}]}, "response": "E[X]=25.20; Var[X]=7.056.", "meta": {"id": "WUA_STAT_25K_013225", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.88, s=26.84, n=70, compute CI."}]}, "response": "95% CI ≈ [18.59, 31.17].", "meta": {"id": "WUA_STAT_25K_013226", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.62, s=25.12, n=168, compute CI."}]}, "response": "95% CI ≈ [194.82, 202.42].", "meta": {"id": "WUA_STAT_25K_013227", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.17, compute E and Var."}]}, "response": "E[X]=4.08; Var[X]=3.386.", "meta": {"id": "WUA_STAT_25K_013228", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.97, μ=14.57, σ=19.06, compute z."}]}, "response": "z = 1.438.", "meta": {"id": "WUA_STAT_25K_013229", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.82, μ=20.84, σ=8.07, compute z."}]}, "response": "z = 8.176.", "meta": {"id": "WUA_STAT_25K_013230", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.78, μ=79.44, σ=17.67, compute z."}]}, "response": "z = -1.509.", "meta": {"id": "WUA_STAT_25K_013231", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.68, μ=59.72, σ=5.05, compute z."}]}, "response": "z = -2.780.", "meta": {"id": "WUA_STAT_25K_013232", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.36, compute E and Var."}]}, "response": "E[X]=3.60; Var[X]=2.304.", "meta": {"id": "WUA_STAT_25K_013233", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.7, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.277.", "meta": {"id": "WUA_STAT_25K_013234", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.39, s=38.57, n=44, compute CI."}]}, "response": "95% CI ≈ [125.99, 148.79].", "meta": {"id": "WUA_STAT_25K_013235", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.99, μ=71.08, σ=23.21, compute z."}]}, "response": "z = -1.641.", "meta": {"id": "WUA_STAT_25K_013236", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.0, s=38.1, n=114, compute CI."}]}, "response": "95% CI ≈ [48.01, 61.99].", "meta": {"id": "WUA_STAT_25K_013237", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.81, s=5.06, n=20, compute CI."}]}, "response": "95% CI ≈ [178.59, 183.03].", "meta": {"id": "WUA_STAT_25K_013238", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.26, compute E and Var."}]}, "response": "E[X]=7.54; Var[X]=5.580.", "meta": {"id": "WUA_STAT_25K_013239", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.72, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.434.", "meta": {"id": "WUA_STAT_25K_013240", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.43, compute E and Var."}]}, "response": "E[X]=15.91; Var[X]=9.069.", "meta": {"id": "WUA_STAT_25K_013241", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.85, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.359.", "meta": {"id": "WUA_STAT_25K_013242", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.08, μ=1.57, σ=10.77, compute z."}]}, "response": "z = 8.404.", "meta": {"id": "WUA_STAT_25K_013243", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.18, μ=27.56, σ=2.24, compute z."}]}, "response": "z = 10.991.", "meta": {"id": "WUA_STAT_25K_013244", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.79, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.043.", "meta": {"id": "WUA_STAT_25K_013245", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.08, μ=13.62, σ=24.98, compute z."}]}, "response": "z = 0.139.", "meta": {"id": "WUA_STAT_25K_013246", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.14, μ=29.65, σ=15.75, compute z."}]}, "response": "z = 3.904.", "meta": {"id": "WUA_STAT_25K_013247", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.57, compute E and Var."}]}, "response": "E[X]=9.69; Var[X]=4.167.", "meta": {"id": "WUA_STAT_25K_013248", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.84, μ=55.29, σ=20.01, compute z."}]}, "response": "z = 0.977.", "meta": {"id": "WUA_STAT_25K_013249", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.75, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.616.", "meta": {"id": "WUA_STAT_25K_013250", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.182, sens=0.78, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.713.", "meta": {"id": "WUA_STAT_25K_013251", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.054, sens=0.85, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.288.", "meta": {"id": "WUA_STAT_25K_013252", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.28, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=4.032.", "meta": {"id": "WUA_STAT_25K_013253", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.81, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.410.", "meta": {"id": "WUA_STAT_25K_013254", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.85, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.312.", "meta": {"id": "WUA_STAT_25K_013255", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.76, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.505.", "meta": {"id": "WUA_STAT_25K_013256", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.62, s=11.35, n=39, compute CI."}]}, "response": "95% CI ≈ [45.06, 52.18].", "meta": {"id": "WUA_STAT_25K_013257", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.97, μ=43.5, σ=14.55, compute z."}]}, "response": "z = -0.861.", "meta": {"id": "WUA_STAT_25K_013258", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.171, sens=0.89, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.754.", "meta": {"id": "WUA_STAT_25K_013259", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.53, μ=4.62, σ=22.97, compute z."}]}, "response": "z = 3.348.", "meta": {"id": "WUA_STAT_25K_013260", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.18, s=16.49, n=62, compute CI."}]}, "response": "95% CI ≈ [152.08, 160.28].", "meta": {"id": "WUA_STAT_25K_013261", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.066, sens=0.71, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.186.", "meta": {"id": "WUA_STAT_25K_013262", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=42.45, s=1.66, n=113, compute CI."}]}, "response": "95% CI ≈ [42.14, 42.76].", "meta": {"id": "WUA_STAT_25K_013263", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.48, μ=43.7, σ=5.96, compute z."}]}, "response": "z = -0.037.", "meta": {"id": "WUA_STAT_25K_013264", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.15, s=18.04, n=41, compute CI."}]}, "response": "95% CI ≈ [119.63, 130.67].", "meta": {"id": "WUA_STAT_25K_013265", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.9, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.682.", "meta": {"id": "WUA_STAT_25K_013266", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.89, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.712.", "meta": {"id": "WUA_STAT_25K_013267", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.93, s=20.93, n=100, compute CI."}]}, "response": "95% CI ≈ [136.83, 145.03].", "meta": {"id": "WUA_STAT_25K_013268", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.84, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.594.", "meta": {"id": "WUA_STAT_25K_013269", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.32, μ=78.58, σ=9.61, compute z."}]}, "response": "z = 0.285.", "meta": {"id": "WUA_STAT_25K_013270", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.234, sens=0.94, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.742.", "meta": {"id": "WUA_STAT_25K_013271", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.48, compute E and Var."}]}, "response": "E[X]=17.28; Var[X]=8.986.", "meta": {"id": "WUA_STAT_25K_013272", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.69, s=13.96, n=10, compute CI."}]}, "response": "95% CI ≈ [14.04, 31.34].", "meta": {"id": "WUA_STAT_25K_013273", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.5, compute E and Var."}]}, "response": "E[X]=3.00; Var[X]=1.500.", "meta": {"id": "WUA_STAT_25K_013274", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=162.29, s=13.18, n=15, compute CI."}]}, "response": "95% CI ≈ [155.62, 168.96].", "meta": {"id": "WUA_STAT_25K_013275", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.13, μ=14.19, σ=6.66, compute z."}]}, "response": "z = 12.604.", "meta": {"id": "WUA_STAT_25K_013276", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.95, μ=50.14, σ=20.77, compute z."}]}, "response": "z = -1.550.", "meta": {"id": "WUA_STAT_25K_013277", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.36, μ=25.5, σ=6.13, compute z."}]}, "response": "z = 3.077.", "meta": {"id": "WUA_STAT_25K_013278", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.1, μ=50.83, σ=2.81, compute z."}]}, "response": "z = 0.808.", "meta": {"id": "WUA_STAT_25K_013279", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.98, s=25.7, n=195, compute CI."}]}, "response": "95% CI ≈ [150.37, 157.59].", "meta": {"id": "WUA_STAT_25K_013280", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.2, s=13.33, n=68, compute CI."}]}, "response": "95% CI ≈ [132.03, 138.37].", "meta": {"id": "WUA_STAT_25K_013281", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.41, μ=11.52, σ=22.91, compute z."}]}, "response": "z = 1.348.", "meta": {"id": "WUA_STAT_25K_013282", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.71, μ=41.08, σ=19.86, compute z."}]}, "response": "z = -0.019.", "meta": {"id": "WUA_STAT_25K_013283", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.49, compute E and Var."}]}, "response": "E[X]=16.66; Var[X]=8.497.", "meta": {"id": "WUA_STAT_25K_013284", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.227, sens=0.81, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.498.", "meta": {"id": "WUA_STAT_25K_013285", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.011, sens=0.99, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.048.", "meta": {"id": "WUA_STAT_25K_013286", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.98, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.480.", "meta": {"id": "WUA_STAT_25K_013287", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.26, μ=78.3, σ=18.99, compute z."}]}, "response": "z = -0.581.", "meta": {"id": "WUA_STAT_25K_013288", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.87, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.414.", "meta": {"id": "WUA_STAT_25K_013289", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.09, compute E and Var."}]}, "response": "E[X]=1.80; Var[X]=1.638.", "meta": {"id": "WUA_STAT_25K_013290", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.9, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.568.", "meta": {"id": "WUA_STAT_25K_013291", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.45, s=19.19, n=137, compute CI."}]}, "response": "95% CI ≈ [195.24, 201.66].", "meta": {"id": "WUA_STAT_25K_013292", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.73, s=9.24, n=200, compute CI."}]}, "response": "95% CI ≈ [39.45, 42.01].", "meta": {"id": "WUA_STAT_25K_013293", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.52, μ=7.3, σ=6.55, compute z."}]}, "response": "z = 9.194.", "meta": {"id": "WUA_STAT_25K_013294", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.47, μ=37.75, σ=13.03, compute z."}]}, "response": "z = 2.204.", "meta": {"id": "WUA_STAT_25K_013295", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.52, compute E and Var."}]}, "response": "E[X]=20.28; Var[X]=9.734.", "meta": {"id": "WUA_STAT_25K_013296", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.55, s=25.36, n=67, compute CI."}]}, "response": "95% CI ≈ [115.48, 127.62].", "meta": {"id": "WUA_STAT_25K_013297", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.029, sens=0.84, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.098.", "meta": {"id": "WUA_STAT_25K_013298", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=147.16, s=20.91, n=85, compute CI."}]}, "response": "95% CI ≈ [142.71, 151.61].", "meta": {"id": "WUA_STAT_25K_013299", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.184, sens=0.93, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.913.", "meta": {"id": "WUA_STAT_25K_013300", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=10.15, s=13.91, n=120, compute CI."}]}, "response": "95% CI ≈ [7.66, 12.64].", "meta": {"id": "WUA_STAT_25K_013301", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.22, s=39.81, n=39, compute CI."}]}, "response": "95% CI ≈ [63.73, 88.71].", "meta": {"id": "WUA_STAT_25K_013302", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=158.76, s=20.26, n=129, compute CI."}]}, "response": "95% CI ≈ [155.26, 162.26].", "meta": {"id": "WUA_STAT_25K_013303", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.68, compute E and Var."}]}, "response": "E[X]=7.48; Var[X]=2.394.", "meta": {"id": "WUA_STAT_25K_013304", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.77, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.258.", "meta": {"id": "WUA_STAT_25K_013305", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.18, compute E and Var."}]}, "response": "E[X]=1.62; Var[X]=1.328.", "meta": {"id": "WUA_STAT_25K_013306", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.77, μ=66.94, σ=8.54, compute z."}]}, "response": "z = -4.938.", "meta": {"id": "WUA_STAT_25K_013307", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.58, μ=0.65, σ=24.97, compute z."}]}, "response": "z = 1.759.", "meta": {"id": "WUA_STAT_25K_013308", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.23, μ=44.0, σ=11.67, compute z."}]}, "response": "z = -2.294.", "meta": {"id": "WUA_STAT_25K_013309", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.66, compute E and Var."}]}, "response": "E[X]=23.76; Var[X]=8.078.", "meta": {"id": "WUA_STAT_25K_013310", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=162.82, s=18.26, n=157, compute CI."}]}, "response": "95% CI ≈ [159.96, 165.68].", "meta": {"id": "WUA_STAT_25K_013311", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.79, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.537.", "meta": {"id": "WUA_STAT_25K_013312", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.98, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.846.", "meta": {"id": "WUA_STAT_25K_013313", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.7, compute E and Var."}]}, "response": "E[X]=11.20; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_013314", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.84, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.805.", "meta": {"id": "WUA_STAT_25K_013315", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.92, μ=77.94, σ=6.04, compute z."}]}, "response": "z = -6.957.", "meta": {"id": "WUA_STAT_25K_013316", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.0, μ=97.66, σ=15.62, compute z."}]}, "response": "z = -0.810.", "meta": {"id": "WUA_STAT_25K_013317", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.209, sens=0.89, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.485.", "meta": {"id": "WUA_STAT_25K_013318", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.09, compute E and Var."}]}, "response": "E[X]=3.42; Var[X]=3.112.", "meta": {"id": "WUA_STAT_25K_013319", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.76, compute E and Var."}]}, "response": "E[X]=24.32; Var[X]=5.837.", "meta": {"id": "WUA_STAT_25K_013320", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.18, μ=42.17, σ=9.2, compute z."}]}, "response": "z = 2.284.", "meta": {"id": "WUA_STAT_25K_013321", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.9, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.366.", "meta": {"id": "WUA_STAT_25K_013322", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.135, sens=0.9, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.326.", "meta": {"id": "WUA_STAT_25K_013323", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.62, s=12.52, n=98, compute CI."}]}, "response": "95% CI ≈ [155.14, 160.10].", "meta": {"id": "WUA_STAT_25K_013324", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.82, s=16.08, n=199, compute CI."}]}, "response": "95% CI ≈ [83.59, 88.05].", "meta": {"id": "WUA_STAT_25K_013325", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.71, μ=64.22, σ=13.01, compute z."}]}, "response": "z = -3.959.", "meta": {"id": "WUA_STAT_25K_013326", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.43, μ=32.39, σ=15.4, compute z."}]}, "response": "z = -1.621.", "meta": {"id": "WUA_STAT_25K_013327", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.104, sens=0.71, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.282.", "meta": {"id": "WUA_STAT_25K_013328", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.99, μ=25.75, σ=7.72, compute z."}]}, "response": "z = 7.285.", "meta": {"id": "WUA_STAT_25K_013329", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.51, s=27.61, n=101, compute CI."}]}, "response": "95% CI ≈ [70.13, 80.89].", "meta": {"id": "WUA_STAT_25K_013330", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.96, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.528.", "meta": {"id": "WUA_STAT_25K_013331", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.207, sens=0.99, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.742.", "meta": {"id": "WUA_STAT_25K_013332", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.52, s=3.93, n=89, compute CI."}]}, "response": "95% CI ≈ [84.70, 86.34].", "meta": {"id": "WUA_STAT_25K_013333", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.38, s=36.45, n=34, compute CI."}]}, "response": "95% CI ≈ [71.13, 95.63].", "meta": {"id": "WUA_STAT_25K_013334", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.51, μ=94.35, σ=10.38, compute z."}]}, "response": "z = -2.778.", "meta": {"id": "WUA_STAT_25K_013335", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.33, compute E and Var."}]}, "response": "E[X]=4.95; Var[X]=3.316.", "meta": {"id": "WUA_STAT_25K_013336", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.162, sens=0.81, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.439.", "meta": {"id": "WUA_STAT_25K_013337", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.95, μ=1.12, σ=13.49, compute z."}]}, "response": "z = 6.659.", "meta": {"id": "WUA_STAT_25K_013338", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.9, μ=18.16, σ=9.65, compute z."}]}, "response": "z = 7.227.", "meta": {"id": "WUA_STAT_25K_013339", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.08, μ=14.15, σ=12.13, compute z."}]}, "response": "z = 6.919.", "meta": {"id": "WUA_STAT_25K_013340", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.71, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.598.", "meta": {"id": "WUA_STAT_25K_013341", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=81.22, s=36.72, n=65, compute CI."}]}, "response": "95% CI ≈ [72.29, 90.15].", "meta": {"id": "WUA_STAT_25K_013342", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.66, compute E and Var."}]}, "response": "E[X]=23.76; Var[X]=8.078.", "meta": {"id": "WUA_STAT_25K_013343", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.38, compute E and Var."}]}, "response": "E[X]=12.92; Var[X]=8.010.", "meta": {"id": "WUA_STAT_25K_013344", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.43, μ=36.69, σ=3.2, compute z."}]}, "response": "z = -4.144.", "meta": {"id": "WUA_STAT_25K_013345", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.73, μ=7.21, σ=4.21, compute z."}]}, "response": "z = 5.349.", "meta": {"id": "WUA_STAT_25K_013346", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.58, s=7.21, n=182, compute CI."}]}, "response": "95% CI ≈ [36.53, 38.63].", "meta": {"id": "WUA_STAT_25K_013347", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.86, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.733.", "meta": {"id": "WUA_STAT_25K_013348", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.11, s=14.43, n=192, compute CI."}]}, "response": "95% CI ≈ [183.07, 187.15].", "meta": {"id": "WUA_STAT_25K_013349", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.68, s=1.8, n=190, compute CI."}]}, "response": "95% CI ≈ [112.42, 112.94].", "meta": {"id": "WUA_STAT_25K_013350", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.69, compute E and Var."}]}, "response": "E[X]=26.22; Var[X]=8.128.", "meta": {"id": "WUA_STAT_25K_013351", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.88, μ=31.8, σ=23.43, compute z."}]}, "response": "z = -0.082.", "meta": {"id": "WUA_STAT_25K_013352", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.69, compute E and Var."}]}, "response": "E[X]=11.73; Var[X]=3.636.", "meta": {"id": "WUA_STAT_25K_013353", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=155.51, s=15.54, n=75, compute CI."}]}, "response": "95% CI ≈ [151.99, 159.03].", "meta": {"id": "WUA_STAT_25K_013354", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.193, sens=0.8, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.502.", "meta": {"id": "WUA_STAT_25K_013355", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.72, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.374.", "meta": {"id": "WUA_STAT_25K_013356", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.75, compute E and Var."}]}, "response": "E[X]=21.00; Var[X]=5.250.", "meta": {"id": "WUA_STAT_25K_013357", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.47, compute E and Var."}]}, "response": "E[X]=8.93; Var[X]=4.733.", "meta": {"id": "WUA_STAT_25K_013358", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.19, μ=58.97, σ=23.87, compute z."}]}, "response": "z = -2.253.", "meta": {"id": "WUA_STAT_25K_013359", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.77, μ=34.92, σ=13.18, compute z."}]}, "response": "z = 0.747.", "meta": {"id": "WUA_STAT_25K_013360", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.13, μ=86.1, σ=5.1, compute z."}]}, "response": "z = -13.720.", "meta": {"id": "WUA_STAT_25K_013361", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.54, compute E and Var."}]}, "response": "E[X]=14.58; Var[X]=6.707.", "meta": {"id": "WUA_STAT_25K_013362", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.79, μ=1.64, σ=4.43, compute z."}]}, "response": "z = 3.871.", "meta": {"id": "WUA_STAT_25K_013363", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.73, compute E and Var."}]}, "response": "E[X]=27.01; Var[X]=7.293.", "meta": {"id": "WUA_STAT_25K_013364", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.84, s=38.03, n=194, compute CI."}]}, "response": "95% CI ≈ [129.49, 140.19].", "meta": {"id": "WUA_STAT_25K_013365", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.31, μ=31.06, σ=5.91, compute z."}]}, "response": "z = -0.465.", "meta": {"id": "WUA_STAT_25K_013366", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.63, compute E and Var."}]}, "response": "E[X]=23.94; Var[X]=8.858.", "meta": {"id": "WUA_STAT_25K_013367", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.17, s=36.39, n=186, compute CI."}]}, "response": "95% CI ≈ [145.94, 156.40].", "meta": {"id": "WUA_STAT_25K_013368", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.67, compute E and Var."}]}, "response": "E[X]=4.02; Var[X]=1.327.", "meta": {"id": "WUA_STAT_25K_013369", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.08, s=37.57, n=36, compute CI."}]}, "response": "95% CI ≈ [73.81, 98.35].", "meta": {"id": "WUA_STAT_25K_013370", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.103, sens=0.85, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.252.", "meta": {"id": "WUA_STAT_25K_013371", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.06, s=13.4, n=54, compute CI."}]}, "response": "95% CI ≈ [72.49, 79.63].", "meta": {"id": "WUA_STAT_25K_013372", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.28, compute E and Var."}]}, "response": "E[X]=10.36; Var[X]=7.459.", "meta": {"id": "WUA_STAT_25K_013373", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.5, compute E and Var."}]}, "response": "E[X]=8.00; Var[X]=4.000.", "meta": {"id": "WUA_STAT_25K_013374", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.63, compute E and Var."}]}, "response": "E[X]=12.60; Var[X]=4.662.", "meta": {"id": "WUA_STAT_25K_013375", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.105, sens=0.71, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.357.", "meta": {"id": "WUA_STAT_25K_013376", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.78, compute E and Var."}]}, "response": "E[X]=14.04; Var[X]=3.089.", "meta": {"id": "WUA_STAT_25K_013377", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.21, sens=0.93, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.692.", "meta": {"id": "WUA_STAT_25K_013378", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.32, compute E and Var."}]}, "response": "E[X]=8.96; Var[X]=6.093.", "meta": {"id": "WUA_STAT_25K_013379", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.78, μ=43.15, σ=23.29, compute z."}]}, "response": "z = 1.830.", "meta": {"id": "WUA_STAT_25K_013380", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.21, compute E and Var."}]}, "response": "E[X]=2.73; Var[X]=2.157.", "meta": {"id": "WUA_STAT_25K_013381", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.42, μ=0.39, σ=5.21, compute z."}]}, "response": "z = 12.866.", "meta": {"id": "WUA_STAT_25K_013382", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.65, compute E and Var."}]}, "response": "E[X]=26.00; Var[X]=9.100.", "meta": {"id": "WUA_STAT_25K_013383", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.68, compute E and Var."}]}, "response": "E[X]=21.76; Var[X]=6.963.", "meta": {"id": "WUA_STAT_25K_013384", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.55, s=11.24, n=18, compute CI."}]}, "response": "95% CI ≈ [43.36, 53.74].", "meta": {"id": "WUA_STAT_25K_013385", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.41, μ=71.51, σ=9.73, compute z."}]}, "response": "z = -0.627.", "meta": {"id": "WUA_STAT_25K_013386", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.08, μ=1.59, σ=7.99, compute z."}]}, "response": "z = 11.075.", "meta": {"id": "WUA_STAT_25K_013387", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.04, μ=86.62, σ=20.28, compute z."}]}, "response": "z = -4.023.", "meta": {"id": "WUA_STAT_25K_013388", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.128, sens=0.78, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.433.", "meta": {"id": "WUA_STAT_25K_013389", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.5, compute E and Var."}]}, "response": "E[X]=8.00; Var[X]=4.000.", "meta": {"id": "WUA_STAT_25K_013390", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.9, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.566.", "meta": {"id": "WUA_STAT_25K_013391", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.58, μ=84.8, σ=17.84, compute z."}]}, "response": "z = -4.553.", "meta": {"id": "WUA_STAT_25K_013392", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=94.79, s=6.96, n=66, compute CI."}]}, "response": "95% CI ≈ [93.11, 96.47].", "meta": {"id": "WUA_STAT_25K_013393", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.79, compute E and Var."}]}, "response": "E[X]=30.81; Var[X]=6.470.", "meta": {"id": "WUA_STAT_25K_013394", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.36, s=39.39, n=111, compute CI."}]}, "response": "95% CI ≈ [18.03, 32.69].", "meta": {"id": "WUA_STAT_25K_013395", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.26, μ=13.14, σ=4.24, compute z."}]}, "response": "z = 13.943.", "meta": {"id": "WUA_STAT_25K_013396", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.78, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.569.", "meta": {"id": "WUA_STAT_25K_013397", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.052, sens=0.73, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.667.", "meta": {"id": "WUA_STAT_25K_013398", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.59, s=29.02, n=107, compute CI."}]}, "response": "95% CI ≈ [18.09, 29.09].", "meta": {"id": "WUA_STAT_25K_013399", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.24, μ=61.96, σ=1.59, compute z."}]}, "response": "z = -36.302.", "meta": {"id": "WUA_STAT_25K_013400", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.37, μ=13.5, σ=22.42, compute z."}]}, "response": "z = 1.912.", "meta": {"id": "WUA_STAT_25K_013401", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.36, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=4.608.", "meta": {"id": "WUA_STAT_25K_013402", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.041, sens=0.78, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.217.", "meta": {"id": "WUA_STAT_25K_013403", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.34, compute E and Var."}]}, "response": "E[X]=6.12; Var[X]=4.039.", "meta": {"id": "WUA_STAT_25K_013404", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.0, μ=37.28, σ=3.34, compute z."}]}, "response": "z = 5.904.", "meta": {"id": "WUA_STAT_25K_013405", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.76, s=33.66, n=193, compute CI."}]}, "response": "95% CI ≈ [131.01, 140.51].", "meta": {"id": "WUA_STAT_25K_013406", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.08, compute E and Var."}]}, "response": "E[X]=2.64; Var[X]=2.429.", "meta": {"id": "WUA_STAT_25K_013407", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.23, μ=42.29, σ=4.78, compute z."}]}, "response": "z = 8.146.", "meta": {"id": "WUA_STAT_25K_013408", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.14, μ=53.83, σ=3.43, compute z."}]}, "response": "z = -1.367.", "meta": {"id": "WUA_STAT_25K_013409", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=20.76, s=12.02, n=86, compute CI."}]}, "response": "95% CI ≈ [18.22, 23.30].", "meta": {"id": "WUA_STAT_25K_013410", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.56, μ=52.58, σ=17.73, compute z."}]}, "response": "z = -2.032.", "meta": {"id": "WUA_STAT_25K_013411", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.81, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.083.", "meta": {"id": "WUA_STAT_25K_013412", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.47, s=5.47, n=27, compute CI."}]}, "response": "95% CI ≈ [11.41, 15.53].", "meta": {"id": "WUA_STAT_25K_013413", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.29, μ=96.98, σ=18.88, compute z."}]}, "response": "z = -1.096.", "meta": {"id": "WUA_STAT_25K_013414", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.05, μ=22.62, σ=13.66, compute z."}]}, "response": "z = 0.031.", "meta": {"id": "WUA_STAT_25K_013415", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.15, s=38.08, n=152, compute CI."}]}, "response": "95% CI ≈ [105.10, 117.20].", "meta": {"id": "WUA_STAT_25K_013416", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.45, s=24.55, n=129, compute CI."}]}, "response": "95% CI ≈ [190.21, 198.69].", "meta": {"id": "WUA_STAT_25K_013417", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.27, compute E and Var."}]}, "response": "E[X]=5.40; Var[X]=3.942.", "meta": {"id": "WUA_STAT_25K_013418", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.87, s=37.97, n=134, compute CI."}]}, "response": "95% CI ≈ [183.44, 196.30].", "meta": {"id": "WUA_STAT_25K_013419", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.2, s=29.47, n=103, compute CI."}]}, "response": "95% CI ≈ [111.51, 122.89].", "meta": {"id": "WUA_STAT_25K_013420", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.09, compute E and Var."}]}, "response": "E[X]=0.63; Var[X]=0.573.", "meta": {"id": "WUA_STAT_25K_013421", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.8, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.612.", "meta": {"id": "WUA_STAT_25K_013422", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=52.64, s=16.22, n=148, compute CI."}]}, "response": "95% CI ≈ [50.03, 55.25].", "meta": {"id": "WUA_STAT_25K_013423", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.33, s=9.38, n=117, compute CI."}]}, "response": "95% CI ≈ [65.63, 69.03].", "meta": {"id": "WUA_STAT_25K_013424", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.09, μ=49.98, σ=17.09, compute z."}]}, "response": "z = -2.744.", "meta": {"id": "WUA_STAT_25K_013425", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.52, compute E and Var."}]}, "response": "E[X]=15.08; Var[X]=7.238.", "meta": {"id": "WUA_STAT_25K_013426", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.79, compute E and Var."}]}, "response": "E[X]=15.80; Var[X]=3.318.", "meta": {"id": "WUA_STAT_25K_013427", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.07, μ=37.52, σ=16.04, compute z."}]}, "response": "z = -1.275.", "meta": {"id": "WUA_STAT_25K_013428", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.06, s=17.45, n=199, compute CI."}]}, "response": "95% CI ≈ [148.64, 153.48].", "meta": {"id": "WUA_STAT_25K_013429", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.3, s=13.69, n=191, compute CI."}]}, "response": "95% CI ≈ [116.36, 120.24].", "meta": {"id": "WUA_STAT_25K_013430", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.2, μ=83.34, σ=19.75, compute z."}]}, "response": "z = -4.058.", "meta": {"id": "WUA_STAT_25K_013431", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.71, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.787.", "meta": {"id": "WUA_STAT_25K_013432", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.34, compute E and Var."}]}, "response": "E[X]=6.12; Var[X]=4.039.", "meta": {"id": "WUA_STAT_25K_013433", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.73, s=31.52, n=147, compute CI."}]}, "response": "95% CI ≈ [41.63, 51.83].", "meta": {"id": "WUA_STAT_25K_013434", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.208, sens=0.76, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.400.", "meta": {"id": "WUA_STAT_25K_013435", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.92, s=8.9, n=131, compute CI."}]}, "response": "95% CI ≈ [36.40, 39.44].", "meta": {"id": "WUA_STAT_25K_013436", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.67, compute E and Var."}]}, "response": "E[X]=17.42; Var[X]=5.749.", "meta": {"id": "WUA_STAT_25K_013437", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.29, μ=21.27, σ=17.89, compute z."}]}, "response": "z = 3.131.", "meta": {"id": "WUA_STAT_25K_013438", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.57, s=8.77, n=27, compute CI."}]}, "response": "95% CI ≈ [8.26, 14.88].", "meta": {"id": "WUA_STAT_25K_013439", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.12, μ=24.04, σ=13.14, compute z."}]}, "response": "z = 0.691.", "meta": {"id": "WUA_STAT_25K_013440", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.52, s=9.27, n=57, compute CI."}]}, "response": "95% CI ≈ [34.11, 38.93].", "meta": {"id": "WUA_STAT_25K_013441", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.78, compute E and Var."}]}, "response": "E[X]=4.68; Var[X]=1.030.", "meta": {"id": "WUA_STAT_25K_013442", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.73, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.117.", "meta": {"id": "WUA_STAT_25K_013443", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.54, μ=14.76, σ=12.56, compute z."}]}, "response": "z = 3.326.", "meta": {"id": "WUA_STAT_25K_013444", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.63, compute E and Var."}]}, "response": "E[X]=10.71; Var[X]=3.963.", "meta": {"id": "WUA_STAT_25K_013445", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=148.01, s=3.38, n=33, compute CI."}]}, "response": "95% CI ≈ [146.86, 149.16].", "meta": {"id": "WUA_STAT_25K_013446", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.65, s=34.11, n=21, compute CI."}]}, "response": "95% CI ≈ [116.06, 145.24].", "meta": {"id": "WUA_STAT_25K_013447", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.88, μ=0.37, σ=15.12, compute z."}]}, "response": "z = 0.695.", "meta": {"id": "WUA_STAT_25K_013448", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.57, μ=34.31, σ=14.99, compute z."}]}, "response": "z = 2.753.", "meta": {"id": "WUA_STAT_25K_013449", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.08, s=30.02, n=25, compute CI."}]}, "response": "95% CI ≈ [97.31, 120.85].", "meta": {"id": "WUA_STAT_25K_013450", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=116.26, s=31.03, n=147, compute CI."}]}, "response": "95% CI ≈ [111.24, 121.28].", "meta": {"id": "WUA_STAT_25K_013451", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.092, sens=0.99, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.715.", "meta": {"id": "WUA_STAT_25K_013452", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.07, compute E and Var."}]}, "response": "E[X]=2.59; Var[X]=2.409.", "meta": {"id": "WUA_STAT_25K_013453", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.247, sens=0.76, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.781.", "meta": {"id": "WUA_STAT_25K_013454", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.69, μ=77.53, σ=11.14, compute z."}]}, "response": "z = -5.192.", "meta": {"id": "WUA_STAT_25K_013455", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.51, μ=58.66, σ=15.48, compute z."}]}, "response": "z = 2.251.", "meta": {"id": "WUA_STAT_25K_013456", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.72, s=8.8, n=173, compute CI."}]}, "response": "95% CI ≈ [16.41, 19.03].", "meta": {"id": "WUA_STAT_25K_013457", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.97, μ=8.56, σ=21.4, compute z."}]}, "response": "z = 2.823.", "meta": {"id": "WUA_STAT_25K_013458", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.38, s=2.34, n=44, compute CI."}]}, "response": "95% CI ≈ [108.69, 110.07].", "meta": {"id": "WUA_STAT_25K_013459", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.62, compute E and Var."}]}, "response": "E[X]=8.06; Var[X]=3.063.", "meta": {"id": "WUA_STAT_25K_013460", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.62, s=33.08, n=177, compute CI."}]}, "response": "95% CI ≈ [172.75, 182.49].", "meta": {"id": "WUA_STAT_25K_013461", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.26, μ=56.24, σ=21.15, compute z."}]}, "response": "z = 0.663.", "meta": {"id": "WUA_STAT_25K_013462", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.17, compute E and Var."}]}, "response": "E[X]=3.74; Var[X]=3.104.", "meta": {"id": "WUA_STAT_25K_013463", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.49, s=25.46, n=45, compute CI."}]}, "response": "95% CI ≈ [4.05, 18.93].", "meta": {"id": "WUA_STAT_25K_013464", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.67, μ=42.03, σ=13.28, compute z."}]}, "response": "z = 2.910.", "meta": {"id": "WUA_STAT_25K_013465", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.96, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.959.", "meta": {"id": "WUA_STAT_25K_013466", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.07, μ=59.03, σ=24.64, compute z."}]}, "response": "z = 1.138.", "meta": {"id": "WUA_STAT_25K_013467", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.09, sens=0.99, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.765.", "meta": {"id": "WUA_STAT_25K_013468", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.67, compute E and Var."}]}, "response": "E[X]=26.80; Var[X]=8.844.", "meta": {"id": "WUA_STAT_25K_013469", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.48, s=4.76, n=67, compute CI."}]}, "response": "95% CI ≈ [89.34, 91.62].", "meta": {"id": "WUA_STAT_25K_013470", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.05, s=5.45, n=143, compute CI."}]}, "response": "95% CI ≈ [100.16, 101.94].", "meta": {"id": "WUA_STAT_25K_013471", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.94, μ=59.37, σ=21.31, compute z."}]}, "response": "z = -0.489.", "meta": {"id": "WUA_STAT_25K_013472", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.28, s=29.99, n=140, compute CI."}]}, "response": "95% CI ≈ [141.31, 151.25].", "meta": {"id": "WUA_STAT_25K_013473", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.24, compute E and Var."}]}, "response": "E[X]=2.64; Var[X]=2.006.", "meta": {"id": "WUA_STAT_25K_013474", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.39, compute E and Var."}]}, "response": "E[X]=4.29; Var[X]=2.617.", "meta": {"id": "WUA_STAT_25K_013475", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.28, μ=95.37, σ=8.24, compute z."}]}, "response": "z = 0.353.", "meta": {"id": "WUA_STAT_25K_013476", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=147.23, s=12.54, n=141, compute CI."}]}, "response": "95% CI ≈ [145.16, 149.30].", "meta": {"id": "WUA_STAT_25K_013477", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.11, compute E and Var."}]}, "response": "E[X]=3.63; Var[X]=3.231.", "meta": {"id": "WUA_STAT_25K_013478", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.71, s=17.2, n=128, compute CI."}]}, "response": "95% CI ≈ [22.73, 28.69].", "meta": {"id": "WUA_STAT_25K_013479", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.72, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.434.", "meta": {"id": "WUA_STAT_25K_013480", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.13, μ=42.63, σ=4.55, compute z."}]}, "response": "z = 7.143.", "meta": {"id": "WUA_STAT_25K_013481", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.08, μ=0.32, σ=13.32, compute z."}]}, "response": "z = 5.462.", "meta": {"id": "WUA_STAT_25K_013482", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.75, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.414.", "meta": {"id": "WUA_STAT_25K_013483", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.19, μ=98.58, σ=12.17, compute z."}]}, "response": "z = -7.509.", "meta": {"id": "WUA_STAT_25K_013484", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.97, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.532.", "meta": {"id": "WUA_STAT_25K_013485", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.086, sens=0.82, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.720.", "meta": {"id": "WUA_STAT_25K_013486", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.17, μ=82.47, σ=11.22, compute z."}]}, "response": "z = 1.399.", "meta": {"id": "WUA_STAT_25K_013487", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.83, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.711.", "meta": {"id": "WUA_STAT_25K_013488", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.72, compute E and Var."}]}, "response": "E[X]=27.36; Var[X]=7.661.", "meta": {"id": "WUA_STAT_25K_013489", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.053, sens=0.77, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.138.", "meta": {"id": "WUA_STAT_25K_013490", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.63, μ=31.42, σ=2.42, compute z."}]}, "response": "z = 5.045.", "meta": {"id": "WUA_STAT_25K_013491", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.6, compute E and Var."}]}, "response": "E[X]=4.20; Var[X]=1.680.", "meta": {"id": "WUA_STAT_25K_013492", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.3, s=14.29, n=148, compute CI."}]}, "response": "95% CI ≈ [104.00, 108.60].", "meta": {"id": "WUA_STAT_25K_013493", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.216, sens=0.91, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.926.", "meta": {"id": "WUA_STAT_25K_013494", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.89, s=27.58, n=39, compute CI."}]}, "response": "95% CI ≈ [127.23, 144.55].", "meta": {"id": "WUA_STAT_25K_013495", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.72, s=2.73, n=66, compute CI."}]}, "response": "95% CI ≈ [166.06, 167.38].", "meta": {"id": "WUA_STAT_25K_013496", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.134, sens=0.8, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.529.", "meta": {"id": "WUA_STAT_25K_013497", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.81, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.190.", "meta": {"id": "WUA_STAT_25K_013498", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=183.75, s=32.56, n=67, compute CI."}]}, "response": "95% CI ≈ [175.95, 191.55].", "meta": {"id": "WUA_STAT_25K_013499", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.97, μ=79.65, σ=22.58, compute z."}]}, "response": "z = 0.634.", "meta": {"id": "WUA_STAT_25K_013500", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=103.3, s=18.52, n=86, compute CI."}]}, "response": "95% CI ≈ [99.39, 107.21].", "meta": {"id": "WUA_STAT_25K_013501", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.82, s=4.17, n=66, compute CI."}]}, "response": "95% CI ≈ [165.81, 167.83].", "meta": {"id": "WUA_STAT_25K_013502", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.65, μ=64.66, σ=20.21, compute z."}]}, "response": "z = -2.178.", "meta": {"id": "WUA_STAT_25K_013503", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.78, compute E and Var."}]}, "response": "E[X]=28.08; Var[X]=6.178.", "meta": {"id": "WUA_STAT_25K_013504", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.3, compute E and Var."}]}, "response": "E[X]=4.50; Var[X]=3.150.", "meta": {"id": "WUA_STAT_25K_013505", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.54, s=24.59, n=145, compute CI."}]}, "response": "95% CI ≈ [116.54, 124.54].", "meta": {"id": "WUA_STAT_25K_013506", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.85, μ=37.85, σ=13.44, compute z."}]}, "response": "z = 4.092.", "meta": {"id": "WUA_STAT_25K_013507", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.25, compute E and Var."}]}, "response": "E[X]=6.50; Var[X]=4.875.", "meta": {"id": "WUA_STAT_25K_013508", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.13, compute E and Var."}]}, "response": "E[X]=3.12; Var[X]=2.714.", "meta": {"id": "WUA_STAT_25K_013509", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.46, μ=34.19, σ=6.85, compute z."}]}, "response": "z = 8.945.", "meta": {"id": "WUA_STAT_25K_013510", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.62, μ=64.26, σ=14.14, compute z."}]}, "response": "z = -2.874.", "meta": {"id": "WUA_STAT_25K_013511", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.29, μ=46.84, σ=1.82, compute z."}]}, "response": "z = -2.500.", "meta": {"id": "WUA_STAT_25K_013512", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.111, sens=0.86, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.264.", "meta": {"id": "WUA_STAT_25K_013513", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.45, μ=75.75, σ=22.06, compute z."}]}, "response": "z = -0.739.", "meta": {"id": "WUA_STAT_25K_013514", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.68, s=21.14, n=17, compute CI."}]}, "response": "95% CI ≈ [134.63, 154.73].", "meta": {"id": "WUA_STAT_25K_013515", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.87, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.386.", "meta": {"id": "WUA_STAT_25K_013516", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.133, sens=0.92, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.402.", "meta": {"id": "WUA_STAT_25K_013517", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.5, μ=56.46, σ=19.63, compute z."}]}, "response": "z = 1.072.", "meta": {"id": "WUA_STAT_25K_013518", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.93, μ=19.34, σ=1.87, compute z."}]}, "response": "z = 31.866.", "meta": {"id": "WUA_STAT_25K_013519", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.95, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.306.", "meta": {"id": "WUA_STAT_25K_013520", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.19, compute E and Var."}]}, "response": "E[X]=0.95; Var[X]=0.769.", "meta": {"id": "WUA_STAT_25K_013521", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.73, μ=13.86, σ=6.6, compute z."}]}, "response": "z = 2.102.", "meta": {"id": "WUA_STAT_25K_013522", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.92, μ=28.27, σ=15.28, compute z."}]}, "response": "z = 3.511.", "meta": {"id": "WUA_STAT_25K_013523", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.35, μ=74.09, σ=10.33, compute z."}]}, "response": "z = -3.266.", "meta": {"id": "WUA_STAT_25K_013524", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=82.82, s=20.27, n=29, compute CI."}]}, "response": "95% CI ≈ [75.44, 90.20].", "meta": {"id": "WUA_STAT_25K_013525", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.067, sens=0.85, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.304.", "meta": {"id": "WUA_STAT_25K_013526", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.207, sens=0.92, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.828.", "meta": {"id": "WUA_STAT_25K_013527", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.54, compute E and Var."}]}, "response": "E[X]=17.28; Var[X]=7.949.", "meta": {"id": "WUA_STAT_25K_013528", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.58, compute E and Var."}]}, "response": "E[X]=5.22; Var[X]=2.192.", "meta": {"id": "WUA_STAT_25K_013529", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.97, s=6.6, n=198, compute CI."}]}, "response": "95% CI ≈ [89.05, 90.89].", "meta": {"id": "WUA_STAT_25K_013530", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.18, μ=82.57, σ=10.86, compute z."}]}, "response": "z = -2.154.", "meta": {"id": "WUA_STAT_25K_013531", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.88, s=34.71, n=104, compute CI."}]}, "response": "95% CI ≈ [71.21, 84.55].", "meta": {"id": "WUA_STAT_25K_013532", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.52, s=32.05, n=59, compute CI."}]}, "response": "95% CI ≈ [93.34, 109.70].", "meta": {"id": "WUA_STAT_25K_013533", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.49, compute E and Var."}]}, "response": "E[X]=7.84; Var[X]=3.998.", "meta": {"id": "WUA_STAT_25K_013534", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.73, μ=21.25, σ=17.86, compute z."}]}, "response": "z = 4.226.", "meta": {"id": "WUA_STAT_25K_013535", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.25, μ=91.76, σ=18.13, compute z."}]}, "response": "z = -1.683.", "meta": {"id": "WUA_STAT_25K_013536", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.024, sens=0.86, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.124.", "meta": {"id": "WUA_STAT_25K_013537", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=192.35, s=7.54, n=135, compute CI."}]}, "response": "95% CI ≈ [191.08, 193.62].", "meta": {"id": "WUA_STAT_25K_013538", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.1, compute E and Var."}]}, "response": "E[X]=1.20; Var[X]=1.080.", "meta": {"id": "WUA_STAT_25K_013539", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.99, μ=31.6, σ=10.92, compute z."}]}, "response": "z = 0.768.", "meta": {"id": "WUA_STAT_25K_013540", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.17, compute E and Var."}]}, "response": "E[X]=3.57; Var[X]=2.963.", "meta": {"id": "WUA_STAT_25K_013541", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.14, compute E and Var."}]}, "response": "E[X]=0.84; Var[X]=0.722.", "meta": {"id": "WUA_STAT_25K_013542", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.224, sens=0.74, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.914.", "meta": {"id": "WUA_STAT_25K_013543", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.88, μ=11.53, σ=17.69, compute z."}]}, "response": "z = 4.316.", "meta": {"id": "WUA_STAT_25K_013544", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.49, compute E and Var."}]}, "response": "E[X]=16.66; Var[X]=8.497.", "meta": {"id": "WUA_STAT_25K_013545", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.86, s=7.29, n=74, compute CI."}]}, "response": "95% CI ≈ [39.20, 42.52].", "meta": {"id": "WUA_STAT_25K_013546", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.81, s=17.96, n=165, compute CI."}]}, "response": "95% CI ≈ [130.07, 135.55].", "meta": {"id": "WUA_STAT_25K_013547", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.66, compute E and Var."}]}, "response": "E[X]=17.82; Var[X]=6.059.", "meta": {"id": "WUA_STAT_25K_013548", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.59, μ=39.11, σ=2.26, compute z."}]}, "response": "z = 20.566.", "meta": {"id": "WUA_STAT_25K_013549", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.58, s=10.97, n=62, compute CI."}]}, "response": "95% CI ≈ [174.85, 180.31].", "meta": {"id": "WUA_STAT_25K_013550", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.207, sens=0.77, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.446.", "meta": {"id": "WUA_STAT_25K_013551", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.75, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.221.", "meta": {"id": "WUA_STAT_25K_013552", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.119, sens=0.86, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.326.", "meta": {"id": "WUA_STAT_25K_013553", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.51, s=33.91, n=104, compute CI."}]}, "response": "95% CI ≈ [180.99, 194.03].", "meta": {"id": "WUA_STAT_25K_013554", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.41, compute E and Var."}]}, "response": "E[X]=8.61; Var[X]=5.080.", "meta": {"id": "WUA_STAT_25K_013555", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.38, compute E and Var."}]}, "response": "E[X]=6.08; Var[X]=3.770.", "meta": {"id": "WUA_STAT_25K_013556", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.31, μ=22.65, σ=7.87, compute z."}]}, "response": "z = 6.945.", "meta": {"id": "WUA_STAT_25K_013557", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.08, μ=7.93, σ=3.74, compute z."}]}, "response": "z = 5.655.", "meta": {"id": "WUA_STAT_25K_013558", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.1, compute E and Var."}]}, "response": "E[X]=2.40; Var[X]=2.160.", "meta": {"id": "WUA_STAT_25K_013559", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.41, compute E and Var."}]}, "response": "E[X]=8.61; Var[X]=5.080.", "meta": {"id": "WUA_STAT_25K_013560", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.06, μ=49.52, σ=15.1, compute z."}]}, "response": "z = 0.102.", "meta": {"id": "WUA_STAT_25K_013561", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.192, sens=0.94, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.598.", "meta": {"id": "WUA_STAT_25K_013562", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.92, μ=67.87, σ=22.71, compute z."}]}, "response": "z = -2.728.", "meta": {"id": "WUA_STAT_25K_013563", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.92, s=38.6, n=147, compute CI."}]}, "response": "95% CI ≈ [171.68, 184.16].", "meta": {"id": "WUA_STAT_25K_013564", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.59, μ=96.0, σ=20.58, compute z."}]}, "response": "z = -0.846.", "meta": {"id": "WUA_STAT_25K_013565", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.89, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.851.", "meta": {"id": "WUA_STAT_25K_013566", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.28, μ=58.88, σ=13.86, compute z."}]}, "response": "z = 2.915.", "meta": {"id": "WUA_STAT_25K_013567", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.106, sens=0.71, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.296.", "meta": {"id": "WUA_STAT_25K_013568", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.2, compute E and Var."}]}, "response": "E[X]=2.60; Var[X]=2.080.", "meta": {"id": "WUA_STAT_25K_013569", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.81, μ=23.58, σ=7.6, compute z."}]}, "response": "z = 4.504.", "meta": {"id": "WUA_STAT_25K_013570", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.59, μ=32.46, σ=9.35, compute z."}]}, "response": "z = 2.474.", "meta": {"id": "WUA_STAT_25K_013571", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.58, μ=65.85, σ=5.32, compute z."}]}, "response": "z = -11.329.", "meta": {"id": "WUA_STAT_25K_013572", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.99, μ=48.59, σ=9.66, compute z."}]}, "response": "z = -4.203.", "meta": {"id": "WUA_STAT_25K_013573", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.05, compute E and Var."}]}, "response": "E[X]=1.80; Var[X]=1.710.", "meta": {"id": "WUA_STAT_25K_013574", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.182, sens=0.72, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.411.", "meta": {"id": "WUA_STAT_25K_013575", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.83, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.331.", "meta": {"id": "WUA_STAT_25K_013576", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.66, μ=65.25, σ=13.34, compute z."}]}, "response": "z = 0.481.", "meta": {"id": "WUA_STAT_25K_013577", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.93, s=3.44, n=124, compute CI."}]}, "response": "95% CI ≈ [112.32, 113.54].", "meta": {"id": "WUA_STAT_25K_013578", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=73.65, s=38.19, n=172, compute CI."}]}, "response": "95% CI ≈ [67.94, 79.36].", "meta": {"id": "WUA_STAT_25K_013579", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.158, sens=0.79, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.832.", "meta": {"id": "WUA_STAT_25K_013580", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.45, compute E and Var."}]}, "response": "E[X]=4.05; Var[X]=2.228.", "meta": {"id": "WUA_STAT_25K_013581", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.218, sens=0.82, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.498.", "meta": {"id": "WUA_STAT_25K_013582", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.23, s=17.94, n=92, compute CI."}]}, "response": "95% CI ≈ [18.56, 25.90].", "meta": {"id": "WUA_STAT_25K_013583", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.48, s=16.84, n=56, compute CI."}]}, "response": "95% CI ≈ [88.07, 96.89].", "meta": {"id": "WUA_STAT_25K_013584", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.99, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.152.", "meta": {"id": "WUA_STAT_25K_013585", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.07, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.953.", "meta": {"id": "WUA_STAT_25K_013586", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.56, μ=10.59, σ=13.8, compute z."}]}, "response": "z = 4.563.", "meta": {"id": "WUA_STAT_25K_013587", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=146.05, s=39.65, n=153, compute CI."}]}, "response": "95% CI ≈ [139.77, 152.33].", "meta": {"id": "WUA_STAT_25K_013588", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.76, μ=22.16, σ=19.73, compute z."}]}, "response": "z = 2.717.", "meta": {"id": "WUA_STAT_25K_013589", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.97, μ=8.65, σ=7.0, compute z."}]}, "response": "z = 4.760.", "meta": {"id": "WUA_STAT_25K_013590", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.159, sens=0.85, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.697.", "meta": {"id": "WUA_STAT_25K_013591", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.57, compute E and Var."}]}, "response": "E[X]=7.41; Var[X]=3.186.", "meta": {"id": "WUA_STAT_25K_013592", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.03, sens=0.73, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.361.", "meta": {"id": "WUA_STAT_25K_013593", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.093, sens=0.78, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.667.", "meta": {"id": "WUA_STAT_25K_013594", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.6, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=2.880.", "meta": {"id": "WUA_STAT_25K_013595", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.95, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.378.", "meta": {"id": "WUA_STAT_25K_013596", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=148.24, s=21.34, n=72, compute CI."}]}, "response": "95% CI ≈ [143.31, 153.17].", "meta": {"id": "WUA_STAT_25K_013597", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.12, sens=0.84, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.342.", "meta": {"id": "WUA_STAT_25K_013598", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.43, μ=44.72, σ=10.12, compute z."}]}, "response": "z = 3.331.", "meta": {"id": "WUA_STAT_25K_013599", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.07, compute E and Var."}]}, "response": "E[X]=2.59; Var[X]=2.409.", "meta": {"id": "WUA_STAT_25K_013600", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.85, s=24.29, n=162, compute CI."}]}, "response": "95% CI ≈ [60.11, 67.59].", "meta": {"id": "WUA_STAT_25K_013601", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.53, compute E and Var."}]}, "response": "E[X]=9.01; Var[X]=4.235.", "meta": {"id": "WUA_STAT_25K_013602", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.06, μ=11.22, σ=11.66, compute z."}]}, "response": "z = 7.019.", "meta": {"id": "WUA_STAT_25K_013603", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.24, compute E and Var."}]}, "response": "E[X]=3.84; Var[X]=2.918.", "meta": {"id": "WUA_STAT_25K_013604", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.7, compute E and Var."}]}, "response": "E[X]=21.00; Var[X]=6.300.", "meta": {"id": "WUA_STAT_25K_013605", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.95, μ=82.88, σ=8.64, compute z."}]}, "response": "z = -8.209.", "meta": {"id": "WUA_STAT_25K_013606", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.24, μ=75.1, σ=19.81, compute z."}]}, "response": "z = -2.164.", "meta": {"id": "WUA_STAT_25K_013607", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=64.07, s=12.4, n=130, compute CI."}]}, "response": "95% CI ≈ [61.94, 66.20].", "meta": {"id": "WUA_STAT_25K_013608", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.06, sens=0.8, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.390.", "meta": {"id": "WUA_STAT_25K_013609", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.187, sens=0.96, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.513.", "meta": {"id": "WUA_STAT_25K_013610", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.73, μ=51.22, σ=6.76, compute z."}]}, "response": "z = 3.330.", "meta": {"id": "WUA_STAT_25K_013611", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.27, compute E and Var."}]}, "response": "E[X]=10.80; Var[X]=7.884.", "meta": {"id": "WUA_STAT_25K_013612", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.5, compute E and Var."}]}, "response": "E[X]=12.50; Var[X]=6.250.", "meta": {"id": "WUA_STAT_25K_013613", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.32, s=16.45, n=152, compute CI."}]}, "response": "95% CI ≈ [116.70, 121.94].", "meta": {"id": "WUA_STAT_25K_013614", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.81, μ=82.12, σ=21.55, compute z."}]}, "response": "z = -2.474.", "meta": {"id": "WUA_STAT_25K_013615", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.054, sens=0.77, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.196.", "meta": {"id": "WUA_STAT_25K_013616", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.97, μ=83.33, σ=4.3, compute z."}]}, "response": "z = -1.712.", "meta": {"id": "WUA_STAT_25K_013617", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.47, s=22.13, n=189, compute CI."}]}, "response": "95% CI ≈ [187.31, 193.63].", "meta": {"id": "WUA_STAT_25K_013618", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.9, μ=65.07, σ=3.11, compute z."}]}, "response": "z = 4.768.", "meta": {"id": "WUA_STAT_25K_013619", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.227, sens=0.91, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.964.", "meta": {"id": "WUA_STAT_25K_013620", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.84, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.092.", "meta": {"id": "WUA_STAT_25K_013621", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=72.73, s=30.68, n=199, compute CI."}]}, "response": "95% CI ≈ [68.47, 76.99].", "meta": {"id": "WUA_STAT_25K_013622", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.82, μ=23.77, σ=4.35, compute z."}]}, "response": "z = 3.460.", "meta": {"id": "WUA_STAT_25K_013623", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.61, s=36.6, n=35, compute CI."}]}, "response": "95% CI ≈ [98.48, 122.74].", "meta": {"id": "WUA_STAT_25K_013624", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.027, sens=0.81, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.273.", "meta": {"id": "WUA_STAT_25K_013625", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.71, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.369.", "meta": {"id": "WUA_STAT_25K_013626", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.2, μ=95.81, σ=14.1, compute z."}]}, "response": "z = -2.667.", "meta": {"id": "WUA_STAT_25K_013627", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.81, s=25.32, n=183, compute CI."}]}, "response": "95% CI ≈ [103.14, 110.48].", "meta": {"id": "WUA_STAT_25K_013628", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.56, compute E and Var."}]}, "response": "E[X]=16.24; Var[X]=7.146.", "meta": {"id": "WUA_STAT_25K_013629", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.78, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.221.", "meta": {"id": "WUA_STAT_25K_013630", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.73, compute E and Var."}]}, "response": "E[X]=22.63; Var[X]=6.110.", "meta": {"id": "WUA_STAT_25K_013631", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.146, sens=0.73, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.609.", "meta": {"id": "WUA_STAT_25K_013632", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.51, compute E and Var."}]}, "response": "E[X]=7.14; Var[X]=3.499.", "meta": {"id": "WUA_STAT_25K_013633", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.84, s=8.28, n=65, compute CI."}]}, "response": "95% CI ≈ [191.83, 195.85].", "meta": {"id": "WUA_STAT_25K_013634", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.35, compute E and Var."}]}, "response": "E[X]=2.45; Var[X]=1.592.", "meta": {"id": "WUA_STAT_25K_013635", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.4, s=3.41, n=143, compute CI."}]}, "response": "95% CI ≈ [26.84, 27.96].", "meta": {"id": "WUA_STAT_25K_013636", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.38, s=35.64, n=106, compute CI."}]}, "response": "95% CI ≈ [187.60, 201.16].", "meta": {"id": "WUA_STAT_25K_013637", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.07, μ=66.49, σ=19.36, compute z."}]}, "response": "z = -1.055.", "meta": {"id": "WUA_STAT_25K_013638", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.09, sens=0.78, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.210.", "meta": {"id": "WUA_STAT_25K_013639", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.049, sens=0.82, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.319.", "meta": {"id": "WUA_STAT_25K_013640", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.46, compute E and Var."}]}, "response": "E[X]=5.06; Var[X]=2.732.", "meta": {"id": "WUA_STAT_25K_013641", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.4, compute E and Var."}]}, "response": "E[X]=3.20; Var[X]=1.920.", "meta": {"id": "WUA_STAT_25K_013642", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.77, μ=79.51, σ=2.33, compute z."}]}, "response": "z = -8.043.", "meta": {"id": "WUA_STAT_25K_013643", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=197.49, s=36.25, n=198, compute CI."}]}, "response": "95% CI ≈ [192.44, 202.54].", "meta": {"id": "WUA_STAT_25K_013644", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.79, compute E and Var."}]}, "response": "E[X]=7.11; Var[X]=1.493.", "meta": {"id": "WUA_STAT_25K_013645", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.83, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.349.", "meta": {"id": "WUA_STAT_25K_013646", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.06, compute E and Var."}]}, "response": "E[X]=2.34; Var[X]=2.200.", "meta": {"id": "WUA_STAT_25K_013647", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.27, s=12.03, n=196, compute CI."}]}, "response": "95% CI ≈ [187.59, 190.95].", "meta": {"id": "WUA_STAT_25K_013648", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.192, sens=0.85, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.437.", "meta": {"id": "WUA_STAT_25K_013649", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.17, s=8.91, n=192, compute CI."}]}, "response": "95% CI ≈ [128.91, 131.43].", "meta": {"id": "WUA_STAT_25K_013650", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.28, compute E and Var."}]}, "response": "E[X]=9.80; Var[X]=7.056.", "meta": {"id": "WUA_STAT_25K_013651", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.95, s=25.87, n=52, compute CI."}]}, "response": "95% CI ≈ [94.92, 108.98].", "meta": {"id": "WUA_STAT_25K_013652", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.0, s=39.94, n=95, compute CI."}]}, "response": "95% CI ≈ [50.97, 67.03].", "meta": {"id": "WUA_STAT_25K_013653", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.89, s=24.53, n=125, compute CI."}]}, "response": "95% CI ≈ [17.59, 26.19].", "meta": {"id": "WUA_STAT_25K_013654", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.87, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.620.", "meta": {"id": "WUA_STAT_25K_013655", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.91, s=20.0, n=95, compute CI."}]}, "response": "95% CI ≈ [181.89, 189.93].", "meta": {"id": "WUA_STAT_25K_013656", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.212, sens=0.86, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.461.", "meta": {"id": "WUA_STAT_25K_013657", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.43, s=30.46, n=115, compute CI."}]}, "response": "95% CI ≈ [70.86, 82.00].", "meta": {"id": "WUA_STAT_25K_013658", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.22, compute E and Var."}]}, "response": "E[X]=7.26; Var[X]=5.663.", "meta": {"id": "WUA_STAT_25K_013659", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.95, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.812.", "meta": {"id": "WUA_STAT_25K_013660", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.89, μ=88.24, σ=16.45, compute z."}]}, "response": "z = -3.365.", "meta": {"id": "WUA_STAT_25K_013661", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.63, s=21.57, n=156, compute CI."}]}, "response": "95% CI ≈ [117.25, 124.01].", "meta": {"id": "WUA_STAT_25K_013662", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.18, compute E and Var."}]}, "response": "E[X]=5.76; Var[X]=4.723.", "meta": {"id": "WUA_STAT_25K_013663", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.93, s=15.23, n=21, compute CI."}]}, "response": "95% CI ≈ [7.42, 20.44].", "meta": {"id": "WUA_STAT_25K_013664", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=81.23, s=28.83, n=163, compute CI."}]}, "response": "95% CI ≈ [76.80, 85.66].", "meta": {"id": "WUA_STAT_25K_013665", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.37, compute E and Var."}]}, "response": "E[X]=14.06; Var[X]=8.858.", "meta": {"id": "WUA_STAT_25K_013666", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.73, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.429.", "meta": {"id": "WUA_STAT_25K_013667", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.56, s=35.35, n=198, compute CI."}]}, "response": "95% CI ≈ [122.64, 132.48].", "meta": {"id": "WUA_STAT_25K_013668", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.8, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.441.", "meta": {"id": "WUA_STAT_25K_013669", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.46, μ=39.86, σ=13.12, compute z."}]}, "response": "z = 3.552.", "meta": {"id": "WUA_STAT_25K_013670", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.7, compute E and Var."}]}, "response": "E[X]=18.20; Var[X]=5.460.", "meta": {"id": "WUA_STAT_25K_013671", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.61, compute E and Var."}]}, "response": "E[X]=20.74; Var[X]=8.089.", "meta": {"id": "WUA_STAT_25K_013672", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.55, compute E and Var."}]}, "response": "E[X]=5.50; Var[X]=2.475.", "meta": {"id": "WUA_STAT_25K_013673", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.29, μ=28.37, σ=9.19, compute z."}]}, "response": "z = -2.620.", "meta": {"id": "WUA_STAT_25K_013674", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.23, compute E and Var."}]}, "response": "E[X]=8.74; Var[X]=6.730.", "meta": {"id": "WUA_STAT_25K_013675", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.86, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.245.", "meta": {"id": "WUA_STAT_25K_013676", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.41, compute E and Var."}]}, "response": "E[X]=10.25; Var[X]=6.048.", "meta": {"id": "WUA_STAT_25K_013677", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.12, s=37.06, n=150, compute CI."}]}, "response": "95% CI ≈ [27.19, 39.05].", "meta": {"id": "WUA_STAT_25K_013678", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.75, compute E and Var."}]}, "response": "E[X]=26.25; Var[X]=6.562.", "meta": {"id": "WUA_STAT_25K_013679", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.35, s=29.24, n=113, compute CI."}]}, "response": "95% CI ≈ [51.96, 62.74].", "meta": {"id": "WUA_STAT_25K_013680", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.89, μ=1.03, σ=22.08, compute z."}]}, "response": "z = 2.938.", "meta": {"id": "WUA_STAT_25K_013681", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.15, μ=93.87, σ=7.76, compute z."}]}, "response": "z = -2.928.", "meta": {"id": "WUA_STAT_25K_013682", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.25, compute E and Var."}]}, "response": "E[X]=3.50; Var[X]=2.625.", "meta": {"id": "WUA_STAT_25K_013683", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.27, s=31.29, n=134, compute CI."}]}, "response": "95% CI ≈ [184.97, 195.57].", "meta": {"id": "WUA_STAT_25K_013684", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.13, s=8.25, n=165, compute CI."}]}, "response": "95% CI ≈ [53.87, 56.39].", "meta": {"id": "WUA_STAT_25K_013685", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.216, sens=0.93, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.631.", "meta": {"id": "WUA_STAT_25K_013686", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.227, sens=0.85, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.893.", "meta": {"id": "WUA_STAT_25K_013687", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.73, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.258.", "meta": {"id": "WUA_STAT_25K_013688", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=158.69, s=10.24, n=16, compute CI."}]}, "response": "95% CI ≈ [153.67, 163.71].", "meta": {"id": "WUA_STAT_25K_013689", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.3, s=11.25, n=90, compute CI."}]}, "response": "95% CI ≈ [176.98, 181.62].", "meta": {"id": "WUA_STAT_25K_013690", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.84, μ=59.22, σ=6.98, compute z."}]}, "response": "z = 4.530.", "meta": {"id": "WUA_STAT_25K_013691", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.43, compute E and Var."}]}, "response": "E[X]=2.15; Var[X]=1.226.", "meta": {"id": "WUA_STAT_25K_013692", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.62, compute E and Var."}]}, "response": "E[X]=16.12; Var[X]=6.126.", "meta": {"id": "WUA_STAT_25K_013693", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.79, compute E and Var."}]}, "response": "E[X]=26.07; Var[X]=5.475.", "meta": {"id": "WUA_STAT_25K_013694", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.78, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.086.", "meta": {"id": "WUA_STAT_25K_013695", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.91, s=23.68, n=23, compute CI."}]}, "response": "95% CI ≈ [124.23, 143.59].", "meta": {"id": "WUA_STAT_25K_013696", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.33, s=18.74, n=82, compute CI."}]}, "response": "95% CI ≈ [42.27, 50.39].", "meta": {"id": "WUA_STAT_25K_013697", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.54, s=3.84, n=103, compute CI."}]}, "response": "95% CI ≈ [150.80, 152.28].", "meta": {"id": "WUA_STAT_25K_013698", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.51, compute E and Var."}]}, "response": "E[X]=8.67; Var[X]=4.248.", "meta": {"id": "WUA_STAT_25K_013699", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.38, s=11.32, n=131, compute CI."}]}, "response": "95% CI ≈ [140.44, 144.32].", "meta": {"id": "WUA_STAT_25K_013700", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.56, compute E and Var."}]}, "response": "E[X]=20.72; Var[X]=9.117.", "meta": {"id": "WUA_STAT_25K_013701", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.18, s=25.32, n=144, compute CI."}]}, "response": "95% CI ≈ [94.04, 102.32].", "meta": {"id": "WUA_STAT_25K_013702", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.69, compute E and Var."}]}, "response": "E[X]=14.49; Var[X]=4.492.", "meta": {"id": "WUA_STAT_25K_013703", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.216, sens=0.74, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.953.", "meta": {"id": "WUA_STAT_25K_013704", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.83, s=12.19, n=178, compute CI."}]}, "response": "95% CI ≈ [106.04, 109.62].", "meta": {"id": "WUA_STAT_25K_013705", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.23, s=11.01, n=111, compute CI."}]}, "response": "95% CI ≈ [20.18, 24.28].", "meta": {"id": "WUA_STAT_25K_013706", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.101, sens=0.84, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.702.", "meta": {"id": "WUA_STAT_25K_013707", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.85, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.668.", "meta": {"id": "WUA_STAT_25K_013708", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.09, μ=22.47, σ=19.32, compute z."}]}, "response": "z = 2.517.", "meta": {"id": "WUA_STAT_25K_013709", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.59, s=6.38, n=51, compute CI."}]}, "response": "95% CI ≈ [178.84, 182.34].", "meta": {"id": "WUA_STAT_25K_013710", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.43, compute E and Var."}]}, "response": "E[X]=3.01; Var[X]=1.716.", "meta": {"id": "WUA_STAT_25K_013711", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.77, compute E and Var."}]}, "response": "E[X]=10.78; Var[X]=2.479.", "meta": {"id": "WUA_STAT_25K_013712", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.82, s=12.75, n=189, compute CI."}]}, "response": "95% CI ≈ [198.00, 201.64].", "meta": {"id": "WUA_STAT_25K_013713", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.82, μ=65.31, σ=4.48, compute z."}]}, "response": "z = 4.132.", "meta": {"id": "WUA_STAT_25K_013714", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.92, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.446.", "meta": {"id": "WUA_STAT_25K_013715", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.85, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.386.", "meta": {"id": "WUA_STAT_25K_013716", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.4, μ=81.63, σ=12.09, compute z."}]}, "response": "z = -6.636.", "meta": {"id": "WUA_STAT_25K_013717", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.69, s=29.03, n=121, compute CI."}]}, "response": "95% CI ≈ [101.52, 111.86].", "meta": {"id": "WUA_STAT_25K_013718", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.64, s=10.44, n=191, compute CI."}]}, "response": "95% CI ≈ [83.16, 86.12].", "meta": {"id": "WUA_STAT_25K_013719", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.51, s=2.02, n=18, compute CI."}]}, "response": "95% CI ≈ [62.58, 64.44].", "meta": {"id": "WUA_STAT_25K_013720", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.96, μ=87.59, σ=13.35, compute z."}]}, "response": "z = -1.021.", "meta": {"id": "WUA_STAT_25K_013721", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.78, s=16.88, n=157, compute CI."}]}, "response": "95% CI ≈ [68.14, 73.42].", "meta": {"id": "WUA_STAT_25K_013722", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.17, compute E and Var."}]}, "response": "E[X]=1.36; Var[X]=1.129.", "meta": {"id": "WUA_STAT_25K_013723", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.96, μ=80.94, σ=3.0, compute z."}]}, "response": "z = -15.327.", "meta": {"id": "WUA_STAT_25K_013724", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.047, sens=0.76, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.294.", "meta": {"id": "WUA_STAT_25K_013725", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.32, compute E and Var."}]}, "response": "E[X]=9.28; Var[X]=6.310.", "meta": {"id": "WUA_STAT_25K_013726", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.32, s=28.42, n=161, compute CI."}]}, "response": "95% CI ≈ [71.93, 80.71].", "meta": {"id": "WUA_STAT_25K_013727", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.06, μ=6.36, σ=6.56, compute z."}]}, "response": "z = 5.137.", "meta": {"id": "WUA_STAT_25K_013728", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.65, s=10.82, n=58, compute CI."}]}, "response": "95% CI ≈ [170.87, 176.43].", "meta": {"id": "WUA_STAT_25K_013729", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.79, compute E and Var."}]}, "response": "E[X]=11.06; Var[X]=2.323.", "meta": {"id": "WUA_STAT_25K_013730", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.74, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.550.", "meta": {"id": "WUA_STAT_25K_013731", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.34, s=33.1, n=96, compute CI."}]}, "response": "95% CI ≈ [53.72, 66.96].", "meta": {"id": "WUA_STAT_25K_013732", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.31, μ=38.98, σ=17.23, compute z."}]}, "response": "z = 0.135.", "meta": {"id": "WUA_STAT_25K_013733", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.044, sens=0.71, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.766.", "meta": {"id": "WUA_STAT_25K_013734", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.24, sens=0.79, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.555.", "meta": {"id": "WUA_STAT_25K_013735", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.84, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.677.", "meta": {"id": "WUA_STAT_25K_013736", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.98, s=16.14, n=163, compute CI."}]}, "response": "95% CI ≈ [179.50, 184.46].", "meta": {"id": "WUA_STAT_25K_013737", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.37, s=28.68, n=134, compute CI."}]}, "response": "95% CI ≈ [84.51, 94.23].", "meta": {"id": "WUA_STAT_25K_013738", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.25, compute E and Var."}]}, "response": "E[X]=10.00; Var[X]=7.500.", "meta": {"id": "WUA_STAT_25K_013739", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.01, s=5.5, n=193, compute CI."}]}, "response": "95% CI ≈ [116.23, 117.79].", "meta": {"id": "WUA_STAT_25K_013740", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.117, sens=0.99, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.384.", "meta": {"id": "WUA_STAT_25K_013741", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.47, s=5.86, n=36, compute CI."}]}, "response": "95% CI ≈ [90.56, 94.38].", "meta": {"id": "WUA_STAT_25K_013742", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.2, sens=0.92, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.821.", "meta": {"id": "WUA_STAT_25K_013743", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.99, μ=55.75, σ=19.46, compute z."}]}, "response": "z = 1.862.", "meta": {"id": "WUA_STAT_25K_013744", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.095, sens=0.88, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.435.", "meta": {"id": "WUA_STAT_25K_013745", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=91.67, s=12.27, n=78, compute CI."}]}, "response": "95% CI ≈ [88.95, 94.39].", "meta": {"id": "WUA_STAT_25K_013746", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.79, μ=31.76, σ=10.6, compute z."}]}, "response": "z = 0.475.", "meta": {"id": "WUA_STAT_25K_013747", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.75, s=26.6, n=111, compute CI."}]}, "response": "95% CI ≈ [145.80, 155.70].", "meta": {"id": "WUA_STAT_25K_013748", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.033, sens=0.91, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.124.", "meta": {"id": "WUA_STAT_25K_013749", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.67, μ=5.34, σ=11.3, compute z."}]}, "response": "z = 3.569.", "meta": {"id": "WUA_STAT_25K_013750", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.05, s=10.36, n=110, compute CI."}]}, "response": "95% CI ≈ [65.11, 68.99].", "meta": {"id": "WUA_STAT_25K_013751", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.221, sens=0.74, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.808.", "meta": {"id": "WUA_STAT_25K_013752", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.85, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.348.", "meta": {"id": "WUA_STAT_25K_013753", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.91, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.280.", "meta": {"id": "WUA_STAT_25K_013754", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.88, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.374.", "meta": {"id": "WUA_STAT_25K_013755", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.15, μ=21.99, σ=13.87, compute z."}]}, "response": "z = 5.130.", "meta": {"id": "WUA_STAT_25K_013756", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.78, μ=44.68, σ=16.34, compute z."}]}, "response": "z = -0.361.", "meta": {"id": "WUA_STAT_25K_013757", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.19, μ=18.32, σ=18.45, compute z."}]}, "response": "z = 4.329.", "meta": {"id": "WUA_STAT_25K_013758", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.2, s=39.36, n=97, compute CI."}]}, "response": "95% CI ≈ [16.37, 32.03].", "meta": {"id": "WUA_STAT_25K_013759", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.08, compute E and Var."}]}, "response": "E[X]=1.12; Var[X]=1.030.", "meta": {"id": "WUA_STAT_25K_013760", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.8, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.235.", "meta": {"id": "WUA_STAT_25K_013761", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.13, μ=25.45, σ=14.19, compute z."}]}, "response": "z = 0.682.", "meta": {"id": "WUA_STAT_25K_013762", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.22, μ=63.86, σ=15.07, compute z."}]}, "response": "z = -2.697.", "meta": {"id": "WUA_STAT_25K_013763", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.25, sens=0.93, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.620.", "meta": {"id": "WUA_STAT_25K_013764", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.75, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.641.", "meta": {"id": "WUA_STAT_25K_013765", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.57, s=24.35, n=95, compute CI."}]}, "response": "95% CI ≈ [69.67, 79.47].", "meta": {"id": "WUA_STAT_25K_013766", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.049, sens=0.78, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.309.", "meta": {"id": "WUA_STAT_25K_013767", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.09, μ=6.46, σ=6.42, compute z."}]}, "response": "z = 10.534.", "meta": {"id": "WUA_STAT_25K_013768", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.61, compute E and Var."}]}, "response": "E[X]=17.69; Var[X]=6.899.", "meta": {"id": "WUA_STAT_25K_013769", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.8, μ=13.0, σ=6.6, compute z."}]}, "response": "z = 13.152.", "meta": {"id": "WUA_STAT_25K_013770", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.185, sens=0.91, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.521.", "meta": {"id": "WUA_STAT_25K_013771", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.8, μ=79.72, σ=4.2, compute z."}]}, "response": "z = 0.495.", "meta": {"id": "WUA_STAT_25K_013772", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.74, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.374.", "meta": {"id": "WUA_STAT_25K_013773", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.45, compute E and Var."}]}, "response": "E[X]=13.95; Var[X]=7.673.", "meta": {"id": "WUA_STAT_25K_013774", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.38, s=11.96, n=168, compute CI."}]}, "response": "95% CI ≈ [10.57, 14.19].", "meta": {"id": "WUA_STAT_25K_013775", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.16, μ=73.06, σ=4.64, compute z."}]}, "response": "z = -12.694.", "meta": {"id": "WUA_STAT_25K_013776", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.81, s=1.03, n=30, compute CI."}]}, "response": "95% CI ≈ [106.44, 107.18].", "meta": {"id": "WUA_STAT_25K_013777", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.146, sens=0.78, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.377.", "meta": {"id": "WUA_STAT_25K_013778", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.99, s=9.79, n=179, compute CI."}]}, "response": "95% CI ≈ [166.56, 169.42].", "meta": {"id": "WUA_STAT_25K_013779", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.92, μ=45.52, σ=12.55, compute z."}]}, "response": "z = 3.139.", "meta": {"id": "WUA_STAT_25K_013780", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.86, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.244.", "meta": {"id": "WUA_STAT_25K_013781", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.224, sens=0.72, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.839.", "meta": {"id": "WUA_STAT_25K_013782", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.74, s=7.45, n=139, compute CI."}]}, "response": "95% CI ≈ [128.50, 130.98].", "meta": {"id": "WUA_STAT_25K_013783", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.65, s=5.13, n=37, compute CI."}]}, "response": "95% CI ≈ [60.00, 63.30].", "meta": {"id": "WUA_STAT_25K_013784", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.87, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.555.", "meta": {"id": "WUA_STAT_25K_013785", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.28, compute E and Var."}]}, "response": "E[X]=3.64; Var[X]=2.621.", "meta": {"id": "WUA_STAT_25K_013786", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.55, s=31.94, n=90, compute CI."}]}, "response": "95% CI ≈ [28.95, 42.15].", "meta": {"id": "WUA_STAT_25K_013787", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.46, compute E and Var."}]}, "response": "E[X]=15.18; Var[X]=8.197.", "meta": {"id": "WUA_STAT_25K_013788", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.72, μ=38.96, σ=3.01, compute z."}]}, "response": "z = -10.711.", "meta": {"id": "WUA_STAT_25K_013789", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.42, μ=44.16, σ=19.02, compute z."}]}, "response": "z = 1.486.", "meta": {"id": "WUA_STAT_25K_013790", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.5, μ=1.19, σ=20.05, compute z."}]}, "response": "z = 3.457.", "meta": {"id": "WUA_STAT_25K_013791", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.69, s=23.88, n=156, compute CI."}]}, "response": "95% CI ≈ [186.94, 194.44].", "meta": {"id": "WUA_STAT_25K_013792", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.62, compute E and Var."}]}, "response": "E[X]=19.84; Var[X]=7.539.", "meta": {"id": "WUA_STAT_25K_013793", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.26, μ=71.65, σ=6.83, compute z."}]}, "response": "z = -7.231.", "meta": {"id": "WUA_STAT_25K_013794", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.18, s=11.56, n=122, compute CI."}]}, "response": "95% CI ≈ [20.13, 24.23].", "meta": {"id": "WUA_STAT_25K_013795", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.81, μ=45.32, σ=13.62, compute z."}]}, "response": "z = 1.504.", "meta": {"id": "WUA_STAT_25K_013796", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.08, s=20.94, n=37, compute CI."}]}, "response": "95% CI ≈ [16.33, 29.83].", "meta": {"id": "WUA_STAT_25K_013797", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.34, compute E and Var."}]}, "response": "E[X]=13.60; Var[X]=8.976.", "meta": {"id": "WUA_STAT_25K_013798", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.7, μ=73.5, σ=4.66, compute z."}]}, "response": "z = -10.472.", "meta": {"id": "WUA_STAT_25K_013799", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.82, s=7.82, n=105, compute CI."}]}, "response": "95% CI ≈ [82.32, 85.32].", "meta": {"id": "WUA_STAT_25K_013800", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=28.43, s=13.53, n=78, compute CI."}]}, "response": "95% CI ≈ [25.43, 31.43].", "meta": {"id": "WUA_STAT_25K_013801", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.07, compute E and Var."}]}, "response": "E[X]=2.45; Var[X]=2.279.", "meta": {"id": "WUA_STAT_25K_013802", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.71, μ=73.76, σ=17.56, compute z."}]}, "response": "z = -1.028.", "meta": {"id": "WUA_STAT_25K_013803", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.157, sens=0.92, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.502.", "meta": {"id": "WUA_STAT_25K_013804", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.56, μ=67.06, σ=8.98, compute z."}]}, "response": "z = -2.506.", "meta": {"id": "WUA_STAT_25K_013805", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.33, compute E and Var."}]}, "response": "E[X]=9.57; Var[X]=6.412.", "meta": {"id": "WUA_STAT_25K_013806", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.98, s=38.35, n=114, compute CI."}]}, "response": "95% CI ≈ [99.94, 114.02].", "meta": {"id": "WUA_STAT_25K_013807", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.89, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.449.", "meta": {"id": "WUA_STAT_25K_013808", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.79, s=34.85, n=24, compute CI."}]}, "response": "95% CI ≈ [47.85, 75.73].", "meta": {"id": "WUA_STAT_25K_013809", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.084, sens=0.93, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.516.", "meta": {"id": "WUA_STAT_25K_013810", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.25, compute E and Var."}]}, "response": "E[X]=10.00; Var[X]=7.500.", "meta": {"id": "WUA_STAT_25K_013811", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.49, compute E and Var."}]}, "response": "E[X]=12.25; Var[X]=6.248.", "meta": {"id": "WUA_STAT_25K_013812", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.72, μ=20.17, σ=6.98, compute z."}]}, "response": "z = -1.497.", "meta": {"id": "WUA_STAT_25K_013813", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.64, μ=20.15, σ=1.58, compute z."}]}, "response": "z = 19.297.", "meta": {"id": "WUA_STAT_25K_013814", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.33, μ=66.7, σ=22.08, compute z."}]}, "response": "z = -0.288.", "meta": {"id": "WUA_STAT_25K_013815", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.67, μ=68.16, σ=22.67, compute z."}]}, "response": "z = -2.315.", "meta": {"id": "WUA_STAT_25K_013816", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.65, compute E and Var."}]}, "response": "E[X]=13.00; Var[X]=4.550.", "meta": {"id": "WUA_STAT_25K_013817", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.29, compute E and Var."}]}, "response": "E[X]=9.28; Var[X]=6.589.", "meta": {"id": "WUA_STAT_25K_013818", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.85, μ=33.24, σ=1.43, compute z."}]}, "response": "z = -12.860.", "meta": {"id": "WUA_STAT_25K_013819", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.062, sens=0.73, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.157.", "meta": {"id": "WUA_STAT_25K_013820", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.71, compute E and Var."}]}, "response": "E[X]=22.01; Var[X]=6.383.", "meta": {"id": "WUA_STAT_25K_013821", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.31, μ=33.17, σ=2.73, compute z."}]}, "response": "z = 8.476.", "meta": {"id": "WUA_STAT_25K_013822", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.88, μ=3.48, σ=9.92, compute z."}]}, "response": "z = 4.980.", "meta": {"id": "WUA_STAT_25K_013823", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.69, compute E and Var."}]}, "response": "E[X]=14.49; Var[X]=4.492.", "meta": {"id": "WUA_STAT_25K_013824", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.233, sens=0.82, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.471.", "meta": {"id": "WUA_STAT_25K_013825", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.57, μ=28.57, σ=3.16, compute z."}]}, "response": "z = 11.709.", "meta": {"id": "WUA_STAT_25K_013826", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.32, μ=92.71, σ=5.25, compute z."}]}, "response": "z = -7.122.", "meta": {"id": "WUA_STAT_25K_013827", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.37, s=6.07, n=175, compute CI."}]}, "response": "95% CI ≈ [73.47, 75.27].", "meta": {"id": "WUA_STAT_25K_013828", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=168.17, s=4.24, n=181, compute CI."}]}, "response": "95% CI ≈ [167.55, 168.79].", "meta": {"id": "WUA_STAT_25K_013829", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.85, s=9.4, n=74, compute CI."}]}, "response": "95% CI ≈ [119.71, 123.99].", "meta": {"id": "WUA_STAT_25K_013830", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.62, s=37.14, n=14, compute CI."}]}, "response": "95% CI ≈ [161.16, 200.08].", "meta": {"id": "WUA_STAT_25K_013831", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.08, s=8.73, n=57, compute CI."}]}, "response": "95% CI ≈ [107.81, 112.35].", "meta": {"id": "WUA_STAT_25K_013832", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.09, s=10.38, n=167, compute CI."}]}, "response": "95% CI ≈ [159.52, 162.66].", "meta": {"id": "WUA_STAT_25K_013833", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.165, sens=0.98, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.418.", "meta": {"id": "WUA_STAT_25K_013834", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.45, compute E and Var."}]}, "response": "E[X]=11.25; Var[X]=6.188.", "meta": {"id": "WUA_STAT_25K_013835", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.18, s=24.78, n=182, compute CI."}]}, "response": "95% CI ≈ [76.58, 83.78].", "meta": {"id": "WUA_STAT_25K_013836", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.64, μ=70.6, σ=5.31, compute z."}]}, "response": "z = -10.350.", "meta": {"id": "WUA_STAT_25K_013837", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.64, compute E and Var."}]}, "response": "E[X]=15.36; Var[X]=5.530.", "meta": {"id": "WUA_STAT_25K_013838", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.85, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.557.", "meta": {"id": "WUA_STAT_25K_013839", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.49, μ=83.51, σ=23.14, compute z."}]}, "response": "z = -3.458.", "meta": {"id": "WUA_STAT_25K_013840", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.8, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.474.", "meta": {"id": "WUA_STAT_25K_013841", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.38, s=11.55, n=109, compute CI."}]}, "response": "95% CI ≈ [137.21, 141.55].", "meta": {"id": "WUA_STAT_25K_013842", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.01, s=4.43, n=38, compute CI."}]}, "response": "95% CI ≈ [36.60, 39.42].", "meta": {"id": "WUA_STAT_25K_013843", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=165.33, s=29.0, n=185, compute CI."}]}, "response": "95% CI ≈ [161.15, 169.51].", "meta": {"id": "WUA_STAT_25K_013844", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.63, compute E and Var."}]}, "response": "E[X]=21.42; Var[X]=7.925.", "meta": {"id": "WUA_STAT_25K_013845", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.25, s=31.4, n=47, compute CI."}]}, "response": "95% CI ≈ [32.27, 50.23].", "meta": {"id": "WUA_STAT_25K_013846", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.4, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=5.040.", "meta": {"id": "WUA_STAT_25K_013847", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.32, compute E and Var."}]}, "response": "E[X]=5.76; Var[X]=3.917.", "meta": {"id": "WUA_STAT_25K_013848", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=148.12, s=3.85, n=110, compute CI."}]}, "response": "95% CI ≈ [147.40, 148.84].", "meta": {"id": "WUA_STAT_25K_013849", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.77, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.451.", "meta": {"id": "WUA_STAT_25K_013850", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.03, sens=0.81, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.111.", "meta": {"id": "WUA_STAT_25K_013851", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.72, s=20.22, n=50, compute CI."}]}, "response": "95% CI ≈ [58.12, 69.32].", "meta": {"id": "WUA_STAT_25K_013852", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.057, sens=0.83, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.626.", "meta": {"id": "WUA_STAT_25K_013853", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.34, s=21.32, n=198, compute CI."}]}, "response": "95% CI ≈ [83.37, 89.31].", "meta": {"id": "WUA_STAT_25K_013854", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.96, μ=17.53, σ=15.63, compute z."}]}, "response": "z = -0.036.", "meta": {"id": "WUA_STAT_25K_013855", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.78, compute E and Var."}]}, "response": "E[X]=7.02; Var[X]=1.544.", "meta": {"id": "WUA_STAT_25K_013856", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.94, s=18.83, n=75, compute CI."}]}, "response": "95% CI ≈ [138.68, 147.20].", "meta": {"id": "WUA_STAT_25K_013857", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.87, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.255.", "meta": {"id": "WUA_STAT_25K_013858", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.47, compute E and Var."}]}, "response": "E[X]=9.40; Var[X]=4.982.", "meta": {"id": "WUA_STAT_25K_013859", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.64, compute E and Var."}]}, "response": "E[X]=19.20; Var[X]=6.912.", "meta": {"id": "WUA_STAT_25K_013860", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.93, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.654.", "meta": {"id": "WUA_STAT_25K_013861", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.104, sens=0.78, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.531.", "meta": {"id": "WUA_STAT_25K_013862", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.55, μ=86.67, σ=5.47, compute z."}]}, "response": "z = -0.753.", "meta": {"id": "WUA_STAT_25K_013863", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=16.05, s=6.65, n=75, compute CI."}]}, "response": "95% CI ≈ [14.54, 17.56].", "meta": {"id": "WUA_STAT_25K_013864", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.73, compute E and Var."}]}, "response": "E[X]=11.68; Var[X]=3.154.", "meta": {"id": "WUA_STAT_25K_013865", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=141.56, s=32.36, n=81, compute CI."}]}, "response": "95% CI ≈ [134.51, 148.61].", "meta": {"id": "WUA_STAT_25K_013866", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.25, compute E and Var."}]}, "response": "E[X]=2.00; Var[X]=1.500.", "meta": {"id": "WUA_STAT_25K_013867", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.71, compute E and Var."}]}, "response": "E[X]=4.97; Var[X]=1.441.", "meta": {"id": "WUA_STAT_25K_013868", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.19, compute E and Var."}]}, "response": "E[X]=6.08; Var[X]=4.925.", "meta": {"id": "WUA_STAT_25K_013869", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.81, s=30.12, n=138, compute CI."}]}, "response": "95% CI ≈ [49.78, 59.84].", "meta": {"id": "WUA_STAT_25K_013870", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.18, compute E and Var."}]}, "response": "E[X]=5.58; Var[X]=4.576.", "meta": {"id": "WUA_STAT_25K_013871", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.55, μ=52.58, σ=14.95, compute z."}]}, "response": "z = 2.807.", "meta": {"id": "WUA_STAT_25K_013872", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.14, s=32.24, n=18, compute CI."}]}, "response": "95% CI ≈ [166.25, 196.03].", "meta": {"id": "WUA_STAT_25K_013873", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.42, compute E and Var."}]}, "response": "E[X]=12.60; Var[X]=7.308.", "meta": {"id": "WUA_STAT_25K_013874", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.46, compute E and Var."}]}, "response": "E[X]=8.74; Var[X]=4.720.", "meta": {"id": "WUA_STAT_25K_013875", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.77, μ=39.34, σ=14.92, compute z."}]}, "response": "z = 1.302.", "meta": {"id": "WUA_STAT_25K_013876", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.69, compute E and Var."}]}, "response": "E[X]=20.70; Var[X]=6.417.", "meta": {"id": "WUA_STAT_25K_013877", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.67, μ=1.5, σ=1.62, compute z."}]}, "response": "z = 58.130.", "meta": {"id": "WUA_STAT_25K_013878", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.88, μ=84.22, σ=4.72, compute z."}]}, "response": "z = -1.131.", "meta": {"id": "WUA_STAT_25K_013879", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.47, μ=52.99, σ=8.9, compute z."}]}, "response": "z = -5.676.", "meta": {"id": "WUA_STAT_25K_013880", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.21, μ=86.78, σ=3.82, compute z."}]}, "response": "z = -11.668.", "meta": {"id": "WUA_STAT_25K_013881", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.89, μ=83.73, σ=1.52, compute z."}]}, "response": "z = -17.000.", "meta": {"id": "WUA_STAT_25K_013882", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.17, compute E and Var."}]}, "response": "E[X]=2.21; Var[X]=1.834.", "meta": {"id": "WUA_STAT_25K_013883", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.74, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.601.", "meta": {"id": "WUA_STAT_25K_013884", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.085, sens=0.83, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.278.", "meta": {"id": "WUA_STAT_25K_013885", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.18, compute E and Var."}]}, "response": "E[X]=5.04; Var[X]=4.133.", "meta": {"id": "WUA_STAT_25K_013886", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.07, compute E and Var."}]}, "response": "E[X]=2.66; Var[X]=2.474.", "meta": {"id": "WUA_STAT_25K_013887", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.71, compute E and Var."}]}, "response": "E[X]=22.01; Var[X]=6.383.", "meta": {"id": "WUA_STAT_25K_013888", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.18, sens=0.85, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.448.", "meta": {"id": "WUA_STAT_25K_013889", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.11, compute E and Var."}]}, "response": "E[X]=3.96; Var[X]=3.524.", "meta": {"id": "WUA_STAT_25K_013890", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.09, compute E and Var."}]}, "response": "E[X]=2.97; Var[X]=2.703.", "meta": {"id": "WUA_STAT_25K_013891", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.47, μ=4.52, σ=4.03, compute z."}]}, "response": "z = 23.065.", "meta": {"id": "WUA_STAT_25K_013892", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.98, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.736.", "meta": {"id": "WUA_STAT_25K_013893", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.75, s=17.2, n=90, compute CI."}]}, "response": "95% CI ≈ [54.20, 61.30].", "meta": {"id": "WUA_STAT_25K_013894", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.74, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.266.", "meta": {"id": "WUA_STAT_25K_013895", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.95, s=5.6, n=170, compute CI."}]}, "response": "95% CI ≈ [117.11, 118.79].", "meta": {"id": "WUA_STAT_25K_013896", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.42, μ=14.39, σ=11.01, compute z."}]}, "response": "z = 3.908.", "meta": {"id": "WUA_STAT_25K_013897", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.8, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.443.", "meta": {"id": "WUA_STAT_25K_013898", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.9, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.363.", "meta": {"id": "WUA_STAT_25K_013899", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.49, compute E and Var."}]}, "response": "E[X]=2.94; Var[X]=1.499.", "meta": {"id": "WUA_STAT_25K_013900", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.8, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.570.", "meta": {"id": "WUA_STAT_25K_013901", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.1, s=31.17, n=179, compute CI."}]}, "response": "95% CI ≈ [74.53, 83.67].", "meta": {"id": "WUA_STAT_25K_013902", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.8, s=39.64, n=164, compute CI."}]}, "response": "95% CI ≈ [29.73, 41.87].", "meta": {"id": "WUA_STAT_25K_013903", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.81, μ=65.14, σ=2.07, compute z."}]}, "response": "z = -27.696.", "meta": {"id": "WUA_STAT_25K_013904", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.68, compute E and Var."}]}, "response": "E[X]=27.20; Var[X]=8.704.", "meta": {"id": "WUA_STAT_25K_013905", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.4, compute E and Var."}]}, "response": "E[X]=14.40; Var[X]=8.640.", "meta": {"id": "WUA_STAT_25K_013906", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.21, s=25.12, n=25, compute CI."}]}, "response": "95% CI ≈ [128.36, 148.06].", "meta": {"id": "WUA_STAT_25K_013907", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.226, sens=0.96, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.849.", "meta": {"id": "WUA_STAT_25K_013908", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.73, compute E and Var."}]}, "response": "E[X]=15.33; Var[X]=4.139.", "meta": {"id": "WUA_STAT_25K_013909", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=155.95, s=16.26, n=20, compute CI."}]}, "response": "95% CI ≈ [148.82, 163.08].", "meta": {"id": "WUA_STAT_25K_013910", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.69, s=33.7, n=43, compute CI."}]}, "response": "95% CI ≈ [11.62, 31.76].", "meta": {"id": "WUA_STAT_25K_013911", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.02, μ=27.42, σ=17.74, compute z."}]}, "response": "z = -0.924.", "meta": {"id": "WUA_STAT_25K_013912", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.107, sens=0.89, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.604.", "meta": {"id": "WUA_STAT_25K_013913", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=47.34, s=36.22, n=185, compute CI."}]}, "response": "95% CI ≈ [42.12, 52.56].", "meta": {"id": "WUA_STAT_25K_013914", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.7, μ=11.21, σ=7.32, compute z."}]}, "response": "z = 9.903.", "meta": {"id": "WUA_STAT_25K_013915", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.55, compute E and Var."}]}, "response": "E[X]=9.35; Var[X]=4.208.", "meta": {"id": "WUA_STAT_25K_013916", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.83, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.586.", "meta": {"id": "WUA_STAT_25K_013917", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.1, compute E and Var."}]}, "response": "E[X]=3.40; Var[X]=3.060.", "meta": {"id": "WUA_STAT_25K_013918", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.119, sens=0.91, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.393.", "meta": {"id": "WUA_STAT_25K_013919", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.32, compute E and Var."}]}, "response": "E[X]=1.60; Var[X]=1.088.", "meta": {"id": "WUA_STAT_25K_013920", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.3, s=11.27, n=97, compute CI."}]}, "response": "95% CI ≈ [99.06, 103.54].", "meta": {"id": "WUA_STAT_25K_013921", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.99, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.950.", "meta": {"id": "WUA_STAT_25K_013922", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.44, μ=69.67, σ=7.27, compute z."}]}, "response": "z = -1.132.", "meta": {"id": "WUA_STAT_25K_013923", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.51, compute E and Var."}]}, "response": "E[X]=15.81; Var[X]=7.747.", "meta": {"id": "WUA_STAT_25K_013924", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.018, sens=0.87, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.615.", "meta": {"id": "WUA_STAT_25K_013925", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.74, s=16.75, n=177, compute CI."}]}, "response": "95% CI ≈ [11.27, 16.21].", "meta": {"id": "WUA_STAT_25K_013926", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.05, compute E and Var."}]}, "response": "E[X]=1.25; Var[X]=1.188.", "meta": {"id": "WUA_STAT_25K_013927", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.75, s=20.55, n=159, compute CI."}]}, "response": "95% CI ≈ [72.56, 78.94].", "meta": {"id": "WUA_STAT_25K_013928", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.0, μ=16.93, σ=2.6, compute z."}]}, "response": "z = 28.488.", "meta": {"id": "WUA_STAT_25K_013929", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.69, compute E and Var."}]}, "response": "E[X]=8.28; Var[X]=2.567.", "meta": {"id": "WUA_STAT_25K_013930", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.76, compute E and Var."}]}, "response": "E[X]=25.08; Var[X]=6.019.", "meta": {"id": "WUA_STAT_25K_013931", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.82, s=14.4, n=32, compute CI."}]}, "response": "95% CI ≈ [29.83, 39.81].", "meta": {"id": "WUA_STAT_25K_013932", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.16, compute E and Var."}]}, "response": "E[X]=1.60; Var[X]=1.344.", "meta": {"id": "WUA_STAT_25K_013933", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.2, μ=40.21, σ=14.47, compute z."}]}, "response": "z = 0.207.", "meta": {"id": "WUA_STAT_25K_013934", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.32, μ=38.49, σ=8.44, compute z."}]}, "response": "z = 5.667.", "meta": {"id": "WUA_STAT_25K_013935", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.015, sens=0.88, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.046.", "meta": {"id": "WUA_STAT_25K_013936", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.67, μ=69.96, σ=17.79, compute z."}]}, "response": "z = -3.726.", "meta": {"id": "WUA_STAT_25K_013937", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.9, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.611.", "meta": {"id": "WUA_STAT_25K_013938", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.95, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.730.", "meta": {"id": "WUA_STAT_25K_013939", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.08, compute E and Var."}]}, "response": "E[X]=0.48; Var[X]=0.442.", "meta": {"id": "WUA_STAT_25K_013940", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.38, compute E and Var."}]}, "response": "E[X]=7.60; Var[X]=4.712.", "meta": {"id": "WUA_STAT_25K_013941", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.99, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.115.", "meta": {"id": "WUA_STAT_25K_013942", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.71, compute E and Var."}]}, "response": "E[X]=19.17; Var[X]=5.559.", "meta": {"id": "WUA_STAT_25K_013943", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.39, compute E and Var."}]}, "response": "E[X]=10.53; Var[X]=6.423.", "meta": {"id": "WUA_STAT_25K_013944", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.62, compute E and Var."}]}, "response": "E[X]=15.50; Var[X]=5.890.", "meta": {"id": "WUA_STAT_25K_013945", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.54, compute E and Var."}]}, "response": "E[X]=18.90; Var[X]=8.694.", "meta": {"id": "WUA_STAT_25K_013946", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.62, compute E and Var."}]}, "response": "E[X]=13.64; Var[X]=5.183.", "meta": {"id": "WUA_STAT_25K_013947", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.097, sens=0.91, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.395.", "meta": {"id": "WUA_STAT_25K_013948", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.2, s=9.21, n=96, compute CI."}]}, "response": "95% CI ≈ [151.36, 155.04].", "meta": {"id": "WUA_STAT_25K_013949", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.81, μ=76.33, σ=18.09, compute z."}]}, "response": "z = 1.243.", "meta": {"id": "WUA_STAT_25K_013950", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.85, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.074.", "meta": {"id": "WUA_STAT_25K_013951", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.92, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.629.", "meta": {"id": "WUA_STAT_25K_013952", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.28, μ=15.5, σ=17.93, compute z."}]}, "response": "z = 3.557.", "meta": {"id": "WUA_STAT_25K_013953", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.29, μ=88.79, σ=4.08, compute z."}]}, "response": "z = -1.838.", "meta": {"id": "WUA_STAT_25K_013954", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.07, compute E and Var."}]}, "response": "E[X]=1.75; Var[X]=1.628.", "meta": {"id": "WUA_STAT_25K_013955", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.8, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.631.", "meta": {"id": "WUA_STAT_25K_013956", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.88, μ=48.08, σ=1.62, compute z."}]}, "response": "z = 8.519.", "meta": {"id": "WUA_STAT_25K_013957", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.5, compute E and Var."}]}, "response": "E[X]=17.50; Var[X]=8.750.", "meta": {"id": "WUA_STAT_25K_013958", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.24, μ=72.03, σ=24.18, compute z."}]}, "response": "z = -2.225.", "meta": {"id": "WUA_STAT_25K_013959", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.29, s=23.13, n=94, compute CI."}]}, "response": "95% CI ≈ [78.61, 87.97].", "meta": {"id": "WUA_STAT_25K_013960", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.097, sens=0.8, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.301.", "meta": {"id": "WUA_STAT_25K_013961", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.221, sens=0.98, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.874.", "meta": {"id": "WUA_STAT_25K_013962", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.42, s=39.8, n=28, compute CI."}]}, "response": "95% CI ≈ [157.68, 187.16].", "meta": {"id": "WUA_STAT_25K_013963", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.29, compute E and Var."}]}, "response": "E[X]=3.48; Var[X]=2.471.", "meta": {"id": "WUA_STAT_25K_013964", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.77, s=3.7, n=17, compute CI."}]}, "response": "95% CI ≈ [49.01, 52.53].", "meta": {"id": "WUA_STAT_25K_013965", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.85, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.448.", "meta": {"id": "WUA_STAT_25K_013966", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.92, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.368.", "meta": {"id": "WUA_STAT_25K_013967", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.31, μ=9.96, σ=6.37, compute z."}]}, "response": "z = 6.648.", "meta": {"id": "WUA_STAT_25K_013968", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.34, compute E and Var."}]}, "response": "E[X]=6.12; Var[X]=4.039.", "meta": {"id": "WUA_STAT_25K_013969", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.09, s=14.02, n=32, compute CI."}]}, "response": "95% CI ≈ [92.23, 101.95].", "meta": {"id": "WUA_STAT_25K_013970", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.26, μ=31.34, σ=9.48, compute z."}]}, "response": "z = -2.329.", "meta": {"id": "WUA_STAT_25K_013971", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.122, sens=0.79, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.785.", "meta": {"id": "WUA_STAT_25K_013972", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.03, s=28.39, n=159, compute CI."}]}, "response": "95% CI ≈ [36.62, 45.44].", "meta": {"id": "WUA_STAT_25K_013973", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.74, s=39.25, n=93, compute CI."}]}, "response": "95% CI ≈ [129.76, 145.72].", "meta": {"id": "WUA_STAT_25K_013974", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.85, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.490.", "meta": {"id": "WUA_STAT_25K_013975", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.73, compute E and Var."}]}, "response": "E[X]=4.38; Var[X]=1.183.", "meta": {"id": "WUA_STAT_25K_013976", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.77, compute E and Var."}]}, "response": "E[X]=26.18; Var[X]=6.021.", "meta": {"id": "WUA_STAT_25K_013977", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.101, sens=0.74, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.257.", "meta": {"id": "WUA_STAT_25K_013978", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.95, μ=64.86, σ=1.59, compute z."}]}, "response": "z = 13.893.", "meta": {"id": "WUA_STAT_25K_013979", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.79, compute E and Var."}]}, "response": "E[X]=27.65; Var[X]=5.806.", "meta": {"id": "WUA_STAT_25K_013980", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=147.6, s=8.08, n=20, compute CI."}]}, "response": "95% CI ≈ [144.06, 151.14].", "meta": {"id": "WUA_STAT_25K_013981", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.43, compute E and Var."}]}, "response": "E[X]=7.31; Var[X]=4.167.", "meta": {"id": "WUA_STAT_25K_013982", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.72, μ=64.28, σ=15.25, compute z."}]}, "response": "z = 1.799.", "meta": {"id": "WUA_STAT_25K_013983", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.23, μ=42.13, σ=8.48, compute z."}]}, "response": "z = -2.347.", "meta": {"id": "WUA_STAT_25K_013984", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.146, sens=0.74, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.644.", "meta": {"id": "WUA_STAT_25K_013985", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.42, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.218.", "meta": {"id": "WUA_STAT_25K_013986", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.62, compute E and Var."}]}, "response": "E[X]=18.60; Var[X]=7.068.", "meta": {"id": "WUA_STAT_25K_013987", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.46, s=38.78, n=75, compute CI."}]}, "response": "95% CI ≈ [68.68, 86.24].", "meta": {"id": "WUA_STAT_25K_013988", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.73, μ=18.27, σ=7.42, compute z."}]}, "response": "z = 2.892.", "meta": {"id": "WUA_STAT_25K_013989", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.057, sens=0.85, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.363.", "meta": {"id": "WUA_STAT_25K_013990", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.29, compute E and Var."}]}, "response": "E[X]=8.99; Var[X]=6.383.", "meta": {"id": "WUA_STAT_25K_013991", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.29, compute E and Var."}]}, "response": "E[X]=8.12; Var[X]=5.765.", "meta": {"id": "WUA_STAT_25K_013992", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.93, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.578.", "meta": {"id": "WUA_STAT_25K_013993", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.018, sens=0.8, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.058.", "meta": {"id": "WUA_STAT_25K_013994", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.6, μ=43.8, σ=16.02, compute z."}]}, "response": "z = 2.547.", "meta": {"id": "WUA_STAT_25K_013995", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.58, compute E and Var."}]}, "response": "E[X]=13.34; Var[X]=5.603.", "meta": {"id": "WUA_STAT_25K_013996", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.55, compute E and Var."}]}, "response": "E[X]=12.10; Var[X]=5.445.", "meta": {"id": "WUA_STAT_25K_013997", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.4, μ=25.95, σ=10.92, compute z."}]}, "response": "z = 1.781.", "meta": {"id": "WUA_STAT_25K_013998", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.55, μ=29.26, σ=16.0, compute z."}]}, "response": "z = -0.232.", "meta": {"id": "WUA_STAT_25K_013999", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.43, compute E and Var."}]}, "response": "E[X]=5.59; Var[X]=3.186.", "meta": {"id": "WUA_STAT_25K_014000", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.44, μ=55.94, σ=16.47, compute z."}]}, "response": "z = 0.941.", "meta": {"id": "WUA_STAT_25K_014001", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.2, compute E and Var."}]}, "response": "E[X]=7.80; Var[X]=6.240.", "meta": {"id": "WUA_STAT_25K_014002", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.77, s=13.19, n=28, compute CI."}]}, "response": "95% CI ≈ [20.88, 30.66].", "meta": {"id": "WUA_STAT_25K_014003", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.72, compute E and Var."}]}, "response": "E[X]=7.92; Var[X]=2.218.", "meta": {"id": "WUA_STAT_25K_014004", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.9, s=31.44, n=98, compute CI."}]}, "response": "95% CI ≈ [47.68, 60.12].", "meta": {"id": "WUA_STAT_25K_014005", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.51, s=36.43, n=85, compute CI."}]}, "response": "95% CI ≈ [113.77, 129.25].", "meta": {"id": "WUA_STAT_25K_014006", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.32, s=37.61, n=162, compute CI."}]}, "response": "95% CI ≈ [130.53, 142.11].", "meta": {"id": "WUA_STAT_25K_014007", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.56, s=38.12, n=173, compute CI."}]}, "response": "95% CI ≈ [187.88, 199.24].", "meta": {"id": "WUA_STAT_25K_014008", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.62, μ=24.75, σ=24.86, compute z."}]}, "response": "z = 0.236.", "meta": {"id": "WUA_STAT_25K_014009", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.75, compute E and Var."}]}, "response": "E[X]=22.50; Var[X]=5.625.", "meta": {"id": "WUA_STAT_25K_014010", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.137, sens=0.94, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.554.", "meta": {"id": "WUA_STAT_25K_014011", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.16, s=20.02, n=91, compute CI."}]}, "response": "95% CI ≈ [136.05, 144.27].", "meta": {"id": "WUA_STAT_25K_014012", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.08, compute E and Var."}]}, "response": "E[X]=2.24; Var[X]=2.061.", "meta": {"id": "WUA_STAT_25K_014013", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.124, sens=0.85, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.354.", "meta": {"id": "WUA_STAT_25K_014014", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.087, sens=0.85, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.323.", "meta": {"id": "WUA_STAT_25K_014015", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.73, μ=24.81, σ=16.19, compute z."}]}, "response": "z = 3.145.", "meta": {"id": "WUA_STAT_25K_014016", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.156, sens=0.73, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.325.", "meta": {"id": "WUA_STAT_25K_014017", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.56, s=19.63, n=199, compute CI."}]}, "response": "95% CI ≈ [171.83, 177.29].", "meta": {"id": "WUA_STAT_25K_014018", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.98, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.084.", "meta": {"id": "WUA_STAT_25K_014019", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.11, compute E and Var."}]}, "response": "E[X]=0.88; Var[X]=0.783.", "meta": {"id": "WUA_STAT_25K_014020", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.1, compute E and Var."}]}, "response": "E[X]=3.40; Var[X]=3.060.", "meta": {"id": "WUA_STAT_25K_014021", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.73, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.498.", "meta": {"id": "WUA_STAT_25K_014022", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.27, compute E and Var."}]}, "response": "E[X]=7.56; Var[X]=5.519.", "meta": {"id": "WUA_STAT_25K_014023", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.175, sens=0.79, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.626.", "meta": {"id": "WUA_STAT_25K_014024", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.4, μ=35.15, σ=15.18, compute z."}]}, "response": "z = 1.927.", "meta": {"id": "WUA_STAT_25K_014025", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.38, compute E and Var."}]}, "response": "E[X]=2.66; Var[X]=1.649.", "meta": {"id": "WUA_STAT_25K_014026", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.115, sens=0.74, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.270.", "meta": {"id": "WUA_STAT_25K_014027", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.19, compute E and Var."}]}, "response": "E[X]=4.56; Var[X]=3.694.", "meta": {"id": "WUA_STAT_25K_014028", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.52, compute E and Var."}]}, "response": "E[X]=20.80; Var[X]=9.984.", "meta": {"id": "WUA_STAT_25K_014029", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.83, μ=54.43, σ=10.27, compute z."}]}, "response": "z = -3.953.", "meta": {"id": "WUA_STAT_25K_014030", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.73, compute E and Var."}]}, "response": "E[X]=15.33; Var[X]=4.139.", "meta": {"id": "WUA_STAT_25K_014031", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.87, μ=96.46, σ=7.42, compute z."}]}, "response": "z = -6.144.", "meta": {"id": "WUA_STAT_25K_014032", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.66, s=1.97, n=84, compute CI."}]}, "response": "95% CI ≈ [80.24, 81.08].", "meta": {"id": "WUA_STAT_25K_014033", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.93, μ=13.01, σ=23.57, compute z."}]}, "response": "z = 0.209.", "meta": {"id": "WUA_STAT_25K_014034", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.39, μ=0.12, σ=21.04, compute z."}]}, "response": "z = 2.674.", "meta": {"id": "WUA_STAT_25K_014035", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.087, sens=0.97, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.270.", "meta": {"id": "WUA_STAT_25K_014036", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.31, s=16.1, n=89, compute CI."}]}, "response": "95% CI ≈ [117.97, 124.65].", "meta": {"id": "WUA_STAT_25K_014037", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=196.91, s=7.35, n=183, compute CI."}]}, "response": "95% CI ≈ [195.85, 197.97].", "meta": {"id": "WUA_STAT_25K_014038", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.225, sens=0.71, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.746.", "meta": {"id": "WUA_STAT_25K_014039", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.76, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.170.", "meta": {"id": "WUA_STAT_25K_014040", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.14, s=19.48, n=83, compute CI."}]}, "response": "95% CI ≈ [151.95, 160.33].", "meta": {"id": "WUA_STAT_25K_014041", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.041, sens=0.79, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.133.", "meta": {"id": "WUA_STAT_25K_014042", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.42, s=23.57, n=90, compute CI."}]}, "response": "95% CI ≈ [87.55, 97.29].", "meta": {"id": "WUA_STAT_25K_014043", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.89, μ=14.59, σ=24.09, compute z."}]}, "response": "z = 2.960.", "meta": {"id": "WUA_STAT_25K_014044", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.92, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.402.", "meta": {"id": "WUA_STAT_25K_014045", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.055, sens=0.91, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.261.", "meta": {"id": "WUA_STAT_25K_014046", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.7, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.261.", "meta": {"id": "WUA_STAT_25K_014047", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.57, compute E and Var."}]}, "response": "E[X]=2.85; Var[X]=1.226.", "meta": {"id": "WUA_STAT_25K_014048", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.72, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.193.", "meta": {"id": "WUA_STAT_25K_014049", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.94, μ=36.68, σ=24.12, compute z."}]}, "response": "z = 2.623.", "meta": {"id": "WUA_STAT_25K_014050", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.05, compute E and Var."}]}, "response": "E[X]=0.60; Var[X]=0.570.", "meta": {"id": "WUA_STAT_25K_014051", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.23, s=15.29, n=168, compute CI."}]}, "response": "95% CI ≈ [77.92, 82.54].", "meta": {"id": "WUA_STAT_25K_014052", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.42, compute E and Var."}]}, "response": "E[X]=7.98; Var[X]=4.628.", "meta": {"id": "WUA_STAT_25K_014053", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=20.7, s=9.67, n=114, compute CI."}]}, "response": "95% CI ≈ [18.92, 22.48].", "meta": {"id": "WUA_STAT_25K_014054", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.5, s=30.34, n=93, compute CI."}]}, "response": "95% CI ≈ [37.33, 49.67].", "meta": {"id": "WUA_STAT_25K_014055", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=71.75, s=32.17, n=199, compute CI."}]}, "response": "95% CI ≈ [67.28, 76.22].", "meta": {"id": "WUA_STAT_25K_014056", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.17, sens=0.88, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.720.", "meta": {"id": "WUA_STAT_25K_014057", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.029, sens=0.88, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.095.", "meta": {"id": "WUA_STAT_25K_014058", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.39, s=20.37, n=82, compute CI."}]}, "response": "95% CI ≈ [182.98, 191.80].", "meta": {"id": "WUA_STAT_25K_014059", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.75, μ=45.57, σ=20.01, compute z."}]}, "response": "z = -1.690.", "meta": {"id": "WUA_STAT_25K_014060", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.84, μ=69.77, σ=6.24, compute z."}]}, "response": "z = 0.011.", "meta": {"id": "WUA_STAT_25K_014061", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.23, s=4.72, n=25, compute CI."}]}, "response": "95% CI ≈ [119.38, 123.08].", "meta": {"id": "WUA_STAT_25K_014062", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.74, compute E and Var."}]}, "response": "E[X]=20.72; Var[X]=5.387.", "meta": {"id": "WUA_STAT_25K_014063", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.33, μ=85.92, σ=21.65, compute z."}]}, "response": "z = -0.628.", "meta": {"id": "WUA_STAT_25K_014064", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.63, s=13.62, n=70, compute CI."}]}, "response": "95% CI ≈ [76.44, 82.82].", "meta": {"id": "WUA_STAT_25K_014065", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.35, s=5.56, n=48, compute CI."}]}, "response": "95% CI ≈ [29.78, 32.92].", "meta": {"id": "WUA_STAT_25K_014066", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.49, compute E and Var."}]}, "response": "E[X]=12.25; Var[X]=6.248.", "meta": {"id": "WUA_STAT_25K_014067", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.124, sens=0.83, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.320.", "meta": {"id": "WUA_STAT_25K_014068", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.32, μ=88.69, σ=14.98, compute z."}]}, "response": "z = -2.361.", "meta": {"id": "WUA_STAT_25K_014069", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.227, sens=0.7, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.495.", "meta": {"id": "WUA_STAT_25K_014070", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=143.59, s=22.73, n=104, compute CI."}]}, "response": "95% CI ≈ [139.22, 147.96].", "meta": {"id": "WUA_STAT_25K_014071", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.23, μ=7.52, σ=7.49, compute z."}]}, "response": "z = 11.844.", "meta": {"id": "WUA_STAT_25K_014072", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.0, s=36.29, n=161, compute CI."}]}, "response": "95% CI ≈ [132.39, 143.61].", "meta": {"id": "WUA_STAT_25K_014073", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.89, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.079.", "meta": {"id": "WUA_STAT_25K_014074", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.36, s=32.57, n=19, compute CI."}]}, "response": "95% CI ≈ [21.71, 51.01].", "meta": {"id": "WUA_STAT_25K_014075", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.43, s=28.9, n=86, compute CI."}]}, "response": "95% CI ≈ [188.32, 200.54].", "meta": {"id": "WUA_STAT_25K_014076", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.24, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=6.384.", "meta": {"id": "WUA_STAT_25K_014077", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.55, compute E and Var."}]}, "response": "E[X]=14.30; Var[X]=6.435.", "meta": {"id": "WUA_STAT_25K_014078", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.9, μ=8.34, σ=22.2, compute z."}]}, "response": "z = 3.178.", "meta": {"id": "WUA_STAT_25K_014079", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.13, s=39.25, n=141, compute CI."}]}, "response": "95% CI ≈ [122.65, 135.61].", "meta": {"id": "WUA_STAT_25K_014080", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.81, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.482.", "meta": {"id": "WUA_STAT_25K_014081", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.03, μ=59.24, σ=17.31, compute z."}]}, "response": "z = 0.219.", "meta": {"id": "WUA_STAT_25K_014082", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.33, s=16.27, n=160, compute CI."}]}, "response": "95% CI ≈ [87.81, 92.85].", "meta": {"id": "WUA_STAT_25K_014083", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.1, compute E and Var."}]}, "response": "E[X]=3.30; Var[X]=2.970.", "meta": {"id": "WUA_STAT_25K_014084", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.86, μ=40.78, σ=5.65, compute z."}]}, "response": "z = 4.970.", "meta": {"id": "WUA_STAT_25K_014085", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.07, μ=65.08, σ=15.05, compute z."}]}, "response": "z = -3.722.", "meta": {"id": "WUA_STAT_25K_014086", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.24, compute E and Var."}]}, "response": "E[X]=6.24; Var[X]=4.742.", "meta": {"id": "WUA_STAT_25K_014087", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.238, sens=0.84, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.621.", "meta": {"id": "WUA_STAT_25K_014088", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.39, compute E and Var."}]}, "response": "E[X]=3.90; Var[X]=2.379.", "meta": {"id": "WUA_STAT_25K_014089", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.91, s=39.07, n=168, compute CI."}]}, "response": "95% CI ≈ [180.00, 191.82].", "meta": {"id": "WUA_STAT_25K_014090", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.81, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.565.", "meta": {"id": "WUA_STAT_25K_014091", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.26, μ=12.53, σ=1.3, compute z."}]}, "response": "z = -3.285.", "meta": {"id": "WUA_STAT_25K_014092", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.34, compute E and Var."}]}, "response": "E[X]=1.70; Var[X]=1.122.", "meta": {"id": "WUA_STAT_25K_014093", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.13, μ=43.8, σ=6.54, compute z."}]}, "response": "z = 2.344.", "meta": {"id": "WUA_STAT_25K_014094", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.3, compute E and Var."}]}, "response": "E[X]=1.50; Var[X]=1.050.", "meta": {"id": "WUA_STAT_25K_014095", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.57, s=33.56, n=188, compute CI."}]}, "response": "95% CI ≈ [32.77, 42.37].", "meta": {"id": "WUA_STAT_25K_014096", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.76, compute E and Var."}]}, "response": "E[X]=22.80; Var[X]=5.472.", "meta": {"id": "WUA_STAT_25K_014097", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.28, s=32.27, n=94, compute CI."}]}, "response": "95% CI ≈ [183.76, 196.80].", "meta": {"id": "WUA_STAT_25K_014098", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=116.28, s=18.7, n=34, compute CI."}]}, "response": "95% CI ≈ [109.99, 122.57].", "meta": {"id": "WUA_STAT_25K_014099", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.05, μ=2.62, σ=1.42, compute z."}]}, "response": "z = 49.599.", "meta": {"id": "WUA_STAT_25K_014100", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.5, s=21.78, n=15, compute CI."}]}, "response": "95% CI ≈ [69.48, 91.52].", "meta": {"id": "WUA_STAT_25K_014101", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.36, μ=52.04, σ=10.9, compute z."}]}, "response": "z = 4.250.", "meta": {"id": "WUA_STAT_25K_014102", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.26, s=33.95, n=152, compute CI."}]}, "response": "95% CI ≈ [145.86, 156.66].", "meta": {"id": "WUA_STAT_25K_014103", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.232, sens=0.92, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.874.", "meta": {"id": "WUA_STAT_25K_014104", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.24, compute E and Var."}]}, "response": "E[X]=6.72; Var[X]=5.107.", "meta": {"id": "WUA_STAT_25K_014105", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.81, μ=70.91, σ=11.61, compute z."}]}, "response": "z = -3.540.", "meta": {"id": "WUA_STAT_25K_014106", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=108.82, s=37.79, n=95, compute CI."}]}, "response": "95% CI ≈ [101.22, 116.42].", "meta": {"id": "WUA_STAT_25K_014107", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.77, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.507.", "meta": {"id": "WUA_STAT_25K_014108", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.99, s=4.57, n=183, compute CI."}]}, "response": "95% CI ≈ [21.33, 22.65].", "meta": {"id": "WUA_STAT_25K_014109", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.8, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.093.", "meta": {"id": "WUA_STAT_25K_014110", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.124, sens=0.86, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.603.", "meta": {"id": "WUA_STAT_25K_014111", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.85, s=29.61, n=176, compute CI."}]}, "response": "95% CI ≈ [63.48, 72.22].", "meta": {"id": "WUA_STAT_25K_014112", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.14, μ=85.23, σ=21.68, compute z."}]}, "response": "z = -3.556.", "meta": {"id": "WUA_STAT_25K_014113", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.48, s=26.39, n=142, compute CI."}]}, "response": "95% CI ≈ [125.14, 133.82].", "meta": {"id": "WUA_STAT_25K_014114", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.48, μ=65.6, σ=20.39, compute z."}]}, "response": "z = 1.073.", "meta": {"id": "WUA_STAT_25K_014115", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.83, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.559.", "meta": {"id": "WUA_STAT_25K_014116", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.08, compute E and Var."}]}, "response": "E[X]=1.44; Var[X]=1.325.", "meta": {"id": "WUA_STAT_25K_014117", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.24, μ=17.95, σ=10.95, compute z."}]}, "response": "z = 5.506.", "meta": {"id": "WUA_STAT_25K_014118", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=93.16, s=19.55, n=169, compute CI."}]}, "response": "95% CI ≈ [90.21, 96.11].", "meta": {"id": "WUA_STAT_25K_014119", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.08, compute E and Var."}]}, "response": "E[X]=3.12; Var[X]=2.870.", "meta": {"id": "WUA_STAT_25K_014120", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.64, compute E and Var."}]}, "response": "E[X]=8.96; Var[X]=3.226.", "meta": {"id": "WUA_STAT_25K_014121", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.59, μ=14.41, σ=1.97, compute z."}]}, "response": "z = 11.259.", "meta": {"id": "WUA_STAT_25K_014122", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.15, compute E and Var."}]}, "response": "E[X]=2.85; Var[X]=2.422.", "meta": {"id": "WUA_STAT_25K_014123", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.4, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=5.040.", "meta": {"id": "WUA_STAT_25K_014124", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.07, μ=16.97, σ=10.95, compute z."}]}, "response": "z = 3.023.", "meta": {"id": "WUA_STAT_25K_014125", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.53, μ=35.86, σ=23.71, compute z."}]}, "response": "z = -0.183.", "meta": {"id": "WUA_STAT_25K_014126", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.97, μ=14.81, σ=8.65, compute z."}]}, "response": "z = 2.215.", "meta": {"id": "WUA_STAT_25K_014127", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.41, s=2.6, n=90, compute CI."}]}, "response": "95% CI ≈ [130.87, 131.95].", "meta": {"id": "WUA_STAT_25K_014128", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.25, compute E and Var."}]}, "response": "E[X]=2.00; Var[X]=1.500.", "meta": {"id": "WUA_STAT_25K_014129", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.69, compute E and Var."}]}, "response": "E[X]=6.21; Var[X]=1.925.", "meta": {"id": "WUA_STAT_25K_014130", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.54, s=33.8, n=117, compute CI."}]}, "response": "95% CI ≈ [9.42, 21.66].", "meta": {"id": "WUA_STAT_25K_014131", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.11, s=35.48, n=128, compute CI."}]}, "response": "95% CI ≈ [17.96, 30.26].", "meta": {"id": "WUA_STAT_25K_014132", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.22, s=8.21, n=21, compute CI."}]}, "response": "95% CI ≈ [76.71, 83.73].", "meta": {"id": "WUA_STAT_25K_014133", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.131, sens=0.78, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.627.", "meta": {"id": "WUA_STAT_25K_014134", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.77, μ=89.75, σ=8.32, compute z."}]}, "response": "z = -5.767.", "meta": {"id": "WUA_STAT_25K_014135", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.81, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.242.", "meta": {"id": "WUA_STAT_25K_014136", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.14, s=33.43, n=146, compute CI."}]}, "response": "95% CI ≈ [148.72, 159.56].", "meta": {"id": "WUA_STAT_25K_014137", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.84, μ=22.73, σ=24.17, compute z."}]}, "response": "z = 1.577.", "meta": {"id": "WUA_STAT_25K_014138", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.3, s=19.73, n=190, compute CI."}]}, "response": "95% CI ≈ [52.49, 58.11].", "meta": {"id": "WUA_STAT_25K_014139", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.41, s=7.61, n=162, compute CI."}]}, "response": "95% CI ≈ [113.24, 115.58].", "meta": {"id": "WUA_STAT_25K_014140", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.08, compute E and Var."}]}, "response": "E[X]=1.04; Var[X]=0.957.", "meta": {"id": "WUA_STAT_25K_014141", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.89, μ=48.27, σ=8.92, compute z."}]}, "response": "z = 3.657.", "meta": {"id": "WUA_STAT_25K_014142", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.43, μ=3.69, σ=13.73, compute z."}]}, "response": "z = 3.768.", "meta": {"id": "WUA_STAT_25K_014143", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.68, s=27.78, n=35, compute CI."}]}, "response": "95% CI ≈ [179.48, 197.88].", "meta": {"id": "WUA_STAT_25K_014144", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.5, compute E and Var."}]}, "response": "E[X]=10.00; Var[X]=5.000.", "meta": {"id": "WUA_STAT_25K_014145", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.19, μ=45.93, σ=4.6, compute z."}]}, "response": "z = 7.883.", "meta": {"id": "WUA_STAT_25K_014146", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.98, μ=18.3, σ=23.01, compute z."}]}, "response": "z = 0.160.", "meta": {"id": "WUA_STAT_25K_014147", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.58, s=11.14, n=100, compute CI."}]}, "response": "95% CI ≈ [117.40, 121.76].", "meta": {"id": "WUA_STAT_25K_014148", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.28, compute E and Var."}]}, "response": "E[X]=5.32; Var[X]=3.830.", "meta": {"id": "WUA_STAT_25K_014149", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.42, s=33.48, n=126, compute CI."}]}, "response": "95% CI ≈ [106.57, 118.27].", "meta": {"id": "WUA_STAT_25K_014150", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.87, μ=76.64, σ=15.85, compute z."}]}, "response": "z = -1.626.", "meta": {"id": "WUA_STAT_25K_014151", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.81, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.330.", "meta": {"id": "WUA_STAT_25K_014152", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.07, s=31.05, n=134, compute CI."}]}, "response": "95% CI ≈ [42.81, 53.33].", "meta": {"id": "WUA_STAT_25K_014153", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.65, s=26.2, n=192, compute CI."}]}, "response": "95% CI ≈ [177.94, 185.36].", "meta": {"id": "WUA_STAT_25K_014154", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=71.4, s=16.38, n=170, compute CI."}]}, "response": "95% CI ≈ [68.94, 73.86].", "meta": {"id": "WUA_STAT_25K_014155", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.82, s=32.41, n=108, compute CI."}]}, "response": "95% CI ≈ [34.71, 46.93].", "meta": {"id": "WUA_STAT_25K_014156", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.65, compute E and Var."}]}, "response": "E[X]=13.65; Var[X]=4.777.", "meta": {"id": "WUA_STAT_25K_014157", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.92, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.474.", "meta": {"id": "WUA_STAT_25K_014158", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.78, μ=13.42, σ=12.66, compute z."}]}, "response": "z = 1.608.", "meta": {"id": "WUA_STAT_25K_014159", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.61, μ=5.12, σ=3.37, compute z."}]}, "response": "z = -0.151.", "meta": {"id": "WUA_STAT_25K_014160", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.26, μ=67.35, σ=10.43, compute z."}]}, "response": "z = -4.994.", "meta": {"id": "WUA_STAT_25K_014161", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.86, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.213.", "meta": {"id": "WUA_STAT_25K_014162", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.28, s=3.7, n=32, compute CI."}]}, "response": "95% CI ≈ [149.00, 151.56].", "meta": {"id": "WUA_STAT_25K_014163", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.212, sens=0.94, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.783.", "meta": {"id": "WUA_STAT_25K_014164", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.041, sens=0.95, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.213.", "meta": {"id": "WUA_STAT_25K_014165", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.73, μ=58.95, σ=15.53, compute z."}]}, "response": "z = -0.722.", "meta": {"id": "WUA_STAT_25K_014166", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.47, compute E and Var."}]}, "response": "E[X]=11.75; Var[X]=6.228.", "meta": {"id": "WUA_STAT_25K_014167", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.77, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.192.", "meta": {"id": "WUA_STAT_25K_014168", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.39, compute E and Var."}]}, "response": "E[X]=4.68; Var[X]=2.855.", "meta": {"id": "WUA_STAT_25K_014169", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.43, compute E and Var."}]}, "response": "E[X]=2.15; Var[X]=1.226.", "meta": {"id": "WUA_STAT_25K_014170", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.14, compute E and Var."}]}, "response": "E[X]=4.20; Var[X]=3.612.", "meta": {"id": "WUA_STAT_25K_014171", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.76, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.260.", "meta": {"id": "WUA_STAT_25K_014172", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.78, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.650.", "meta": {"id": "WUA_STAT_25K_014173", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.015, sens=0.97, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.050.", "meta": {"id": "WUA_STAT_25K_014174", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=169.82, s=38.44, n=31, compute CI."}]}, "response": "95% CI ≈ [156.29, 183.35].", "meta": {"id": "WUA_STAT_25K_014175", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.51, μ=77.26, σ=3.63, compute z."}]}, "response": "z = -12.328.", "meta": {"id": "WUA_STAT_25K_014176", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.95, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.220.", "meta": {"id": "WUA_STAT_25K_014177", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.018, sens=0.8, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.068.", "meta": {"id": "WUA_STAT_25K_014178", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.91, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.959.", "meta": {"id": "WUA_STAT_25K_014179", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.67, compute E and Var."}]}, "response": "E[X]=20.77; Var[X]=6.854.", "meta": {"id": "WUA_STAT_25K_014180", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.14, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=2.167.", "meta": {"id": "WUA_STAT_25K_014181", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.68, μ=88.88, σ=5.16, compute z."}]}, "response": "z = -4.496.", "meta": {"id": "WUA_STAT_25K_014182", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.78, compute E and Var."}]}, "response": "E[X]=29.64; Var[X]=6.521.", "meta": {"id": "WUA_STAT_25K_014183", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.59, μ=87.62, σ=12.68, compute z."}]}, "response": "z = -3.315.", "meta": {"id": "WUA_STAT_25K_014184", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.75, compute E and Var."}]}, "response": "E[X]=23.25; Var[X]=5.812.", "meta": {"id": "WUA_STAT_25K_014185", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.86, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.855.", "meta": {"id": "WUA_STAT_25K_014186", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.119, sens=0.85, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.377.", "meta": {"id": "WUA_STAT_25K_014187", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.69, s=24.8, n=64, compute CI."}]}, "response": "95% CI ≈ [126.61, 138.77].", "meta": {"id": "WUA_STAT_25K_014188", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.124, sens=0.78, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.315.", "meta": {"id": "WUA_STAT_25K_014189", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.44, s=2.1, n=85, compute CI."}]}, "response": "95% CI ≈ [59.99, 60.89].", "meta": {"id": "WUA_STAT_25K_014190", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.09, compute E and Var."}]}, "response": "E[X]=1.89; Var[X]=1.720.", "meta": {"id": "WUA_STAT_25K_014191", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.63, μ=70.94, σ=1.56, compute z."}]}, "response": "z = -43.788.", "meta": {"id": "WUA_STAT_25K_014192", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.13, μ=34.5, σ=6.13, compute z."}]}, "response": "z = 6.954.", "meta": {"id": "WUA_STAT_25K_014193", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.68, s=25.52, n=27, compute CI."}]}, "response": "95% CI ≈ [40.05, 59.31].", "meta": {"id": "WUA_STAT_25K_014194", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.213, sens=0.89, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.572.", "meta": {"id": "WUA_STAT_25K_014195", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.42, μ=41.78, σ=11.55, compute z."}]}, "response": "z = 4.211.", "meta": {"id": "WUA_STAT_25K_014196", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=143.14, s=23.5, n=181, compute CI."}]}, "response": "95% CI ≈ [139.72, 146.56].", "meta": {"id": "WUA_STAT_25K_014197", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.171, sens=0.91, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.419.", "meta": {"id": "WUA_STAT_25K_014198", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.66, μ=92.67, σ=8.65, compute z."}]}, "response": "z = -7.862.", "meta": {"id": "WUA_STAT_25K_014199", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.95, s=15.7, n=120, compute CI."}]}, "response": "95% CI ≈ [29.14, 34.76].", "meta": {"id": "WUA_STAT_25K_014200", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.59, compute E and Var."}]}, "response": "E[X]=4.72; Var[X]=1.935.", "meta": {"id": "WUA_STAT_25K_014201", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.54, μ=47.43, σ=10.82, compute z."}]}, "response": "z = -0.452.", "meta": {"id": "WUA_STAT_25K_014202", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.096, sens=0.89, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.825.", "meta": {"id": "WUA_STAT_25K_014203", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.11, μ=57.32, σ=3.99, compute z."}]}, "response": "z = 10.474.", "meta": {"id": "WUA_STAT_25K_014204", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.42, μ=59.86, σ=3.0, compute z."}]}, "response": "z = -6.480.", "meta": {"id": "WUA_STAT_25K_014205", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.027, sens=0.78, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.086.", "meta": {"id": "WUA_STAT_25K_014206", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.39, compute E and Var."}]}, "response": "E[X]=2.73; Var[X]=1.665.", "meta": {"id": "WUA_STAT_25K_014207", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.05, s=20.48, n=110, compute CI."}]}, "response": "95% CI ≈ [110.22, 117.88].", "meta": {"id": "WUA_STAT_25K_014208", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.046, sens=0.79, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.120.", "meta": {"id": "WUA_STAT_25K_014209", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=73.28, s=12.47, n=134, compute CI."}]}, "response": "95% CI ≈ [71.17, 75.39].", "meta": {"id": "WUA_STAT_25K_014210", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.68, μ=28.35, σ=22.37, compute z."}]}, "response": "z = -1.148.", "meta": {"id": "WUA_STAT_25K_014211", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.73, compute E and Var."}]}, "response": "E[X]=24.82; Var[X]=6.701.", "meta": {"id": "WUA_STAT_25K_014212", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.77, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.620.", "meta": {"id": "WUA_STAT_25K_014213", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.79, compute E and Var."}]}, "response": "E[X]=12.64; Var[X]=2.654.", "meta": {"id": "WUA_STAT_25K_014214", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.96, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.576.", "meta": {"id": "WUA_STAT_25K_014215", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.28, μ=39.61, σ=18.71, compute z."}]}, "response": "z = -1.461.", "meta": {"id": "WUA_STAT_25K_014216", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.38, compute E and Var."}]}, "response": "E[X]=6.46; Var[X]=4.005.", "meta": {"id": "WUA_STAT_25K_014217", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.097, sens=0.77, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.256.", "meta": {"id": "WUA_STAT_25K_014218", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=165.62, s=16.35, n=98, compute CI."}]}, "response": "95% CI ≈ [162.38, 168.86].", "meta": {"id": "WUA_STAT_25K_014219", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.27, s=31.9, n=105, compute CI."}]}, "response": "95% CI ≈ [74.17, 86.37].", "meta": {"id": "WUA_STAT_25K_014220", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.42, compute E and Var."}]}, "response": "E[X]=7.98; Var[X]=4.628.", "meta": {"id": "WUA_STAT_25K_014221", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.21, s=31.76, n=35, compute CI."}]}, "response": "95% CI ≈ [0.69, 21.73].", "meta": {"id": "WUA_STAT_25K_014222", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.74, compute E and Var."}]}, "response": "E[X]=28.12; Var[X]=7.311.", "meta": {"id": "WUA_STAT_25K_014223", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.028, sens=0.85, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.182.", "meta": {"id": "WUA_STAT_25K_014224", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.14, compute E and Var."}]}, "response": "E[X]=5.46; Var[X]=4.696.", "meta": {"id": "WUA_STAT_25K_014225", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.68, μ=45.46, σ=11.1, compute z."}]}, "response": "z = 1.011.", "meta": {"id": "WUA_STAT_25K_014226", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.148, sens=0.85, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.532.", "meta": {"id": "WUA_STAT_25K_014227", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.94, μ=29.39, σ=24.11, compute z."}]}, "response": "z = 0.852.", "meta": {"id": "WUA_STAT_25K_014228", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.39, compute E and Var."}]}, "response": "E[X]=6.63; Var[X]=4.044.", "meta": {"id": "WUA_STAT_25K_014229", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.05, μ=82.92, σ=22.83, compute z."}]}, "response": "z = -3.455.", "meta": {"id": "WUA_STAT_25K_014230", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.108, sens=0.94, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.655.", "meta": {"id": "WUA_STAT_25K_014231", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.78, compute E and Var."}]}, "response": "E[X]=11.70; Var[X]=2.574.", "meta": {"id": "WUA_STAT_25K_014232", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.87, s=21.85, n=77, compute CI."}]}, "response": "95% CI ≈ [104.99, 114.75].", "meta": {"id": "WUA_STAT_25K_014233", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.14, compute E and Var."}]}, "response": "E[X]=2.80; Var[X]=2.408.", "meta": {"id": "WUA_STAT_25K_014234", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.23, μ=24.52, σ=24.4, compute z."}]}, "response": "z = 2.078.", "meta": {"id": "WUA_STAT_25K_014235", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.78, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.134.", "meta": {"id": "WUA_STAT_25K_014236", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.66, μ=73.67, σ=4.34, compute z."}]}, "response": "z = -7.145.", "meta": {"id": "WUA_STAT_25K_014237", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.217, sens=0.73, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.648.", "meta": {"id": "WUA_STAT_25K_014238", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.69, s=12.45, n=21, compute CI."}]}, "response": "95% CI ≈ [18.37, 29.01].", "meta": {"id": "WUA_STAT_25K_014239", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.7, μ=40.6, σ=19.9, compute z."}]}, "response": "z = 1.462.", "meta": {"id": "WUA_STAT_25K_014240", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.78, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.546.", "meta": {"id": "WUA_STAT_25K_014241", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.91, μ=76.23, σ=5.14, compute z."}]}, "response": "z = -0.451.", "meta": {"id": "WUA_STAT_25K_014242", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.98, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.815.", "meta": {"id": "WUA_STAT_25K_014243", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.8, μ=59.34, σ=20.19, compute z."}]}, "response": "z = -2.404.", "meta": {"id": "WUA_STAT_25K_014244", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.226, sens=0.86, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.758.", "meta": {"id": "WUA_STAT_25K_014245", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.81, s=3.38, n=62, compute CI."}]}, "response": "95% CI ≈ [132.97, 134.65].", "meta": {"id": "WUA_STAT_25K_014246", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.63, s=12.85, n=56, compute CI."}]}, "response": "95% CI ≈ [42.26, 49.00].", "meta": {"id": "WUA_STAT_25K_014247", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.8, μ=34.68, σ=6.35, compute z."}]}, "response": "z = 5.846.", "meta": {"id": "WUA_STAT_25K_014248", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.79, s=15.31, n=157, compute CI."}]}, "response": "95% CI ≈ [191.40, 196.18].", "meta": {"id": "WUA_STAT_25K_014249", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.046, sens=0.93, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.169.", "meta": {"id": "WUA_STAT_25K_014250", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.85, s=29.33, n=182, compute CI."}]}, "response": "95% CI ≈ [130.59, 139.11].", "meta": {"id": "WUA_STAT_25K_014251", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=19.94, s=17.78, n=42, compute CI."}]}, "response": "95% CI ≈ [14.56, 25.32].", "meta": {"id": "WUA_STAT_25K_014252", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.54, μ=83.52, σ=11.11, compute z."}]}, "response": "z = -3.959.", "meta": {"id": "WUA_STAT_25K_014253", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.68, μ=71.37, σ=15.0, compute z."}]}, "response": "z = 0.687.", "meta": {"id": "WUA_STAT_25K_014254", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.63, μ=26.83, σ=12.3, compute z."}]}, "response": "z = 2.423.", "meta": {"id": "WUA_STAT_25K_014255", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.18, s=7.83, n=167, compute CI."}]}, "response": "95% CI ≈ [36.99, 39.37].", "meta": {"id": "WUA_STAT_25K_014256", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.084, sens=0.76, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.279.", "meta": {"id": "WUA_STAT_25K_014257", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.87, s=33.15, n=46, compute CI."}]}, "response": "95% CI ≈ [154.29, 173.45].", "meta": {"id": "WUA_STAT_25K_014258", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.0, μ=13.84, σ=3.2, compute z."}]}, "response": "z = 3.487.", "meta": {"id": "WUA_STAT_25K_014259", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.9, s=14.38, n=25, compute CI."}]}, "response": "95% CI ≈ [180.26, 191.54].", "meta": {"id": "WUA_STAT_25K_014260", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.37, μ=0.68, σ=20.98, compute z."}]}, "response": "z = 3.941.", "meta": {"id": "WUA_STAT_25K_014261", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.05, sens=0.77, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.168.", "meta": {"id": "WUA_STAT_25K_014262", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.32, compute E and Var."}]}, "response": "E[X]=3.84; Var[X]=2.611.", "meta": {"id": "WUA_STAT_25K_014263", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.24, s=25.99, n=51, compute CI."}]}, "response": "95% CI ≈ [179.11, 193.37].", "meta": {"id": "WUA_STAT_25K_014264", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.36, μ=57.21, σ=6.69, compute z."}]}, "response": "z = 6.151.", "meta": {"id": "WUA_STAT_25K_014265", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.011, sens=0.9, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.040.", "meta": {"id": "WUA_STAT_25K_014266", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.67, s=11.05, n=84, compute CI."}]}, "response": "95% CI ≈ [96.31, 101.03].", "meta": {"id": "WUA_STAT_25K_014267", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.2, μ=39.59, σ=24.57, compute z."}]}, "response": "z = 2.101.", "meta": {"id": "WUA_STAT_25K_014268", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.79, s=15.86, n=184, compute CI."}]}, "response": "95% CI ≈ [187.50, 192.08].", "meta": {"id": "WUA_STAT_25K_014269", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.64, μ=46.75, σ=19.83, compute z."}]}, "response": "z = 1.507.", "meta": {"id": "WUA_STAT_25K_014270", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.94, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.243.", "meta": {"id": "WUA_STAT_25K_014271", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.45, compute E and Var."}]}, "response": "E[X]=8.55; Var[X]=4.703.", "meta": {"id": "WUA_STAT_25K_014272", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.76, μ=53.17, σ=5.64, compute z."}]}, "response": "z = 5.956.", "meta": {"id": "WUA_STAT_25K_014273", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.09, s=33.89, n=163, compute CI."}]}, "response": "95% CI ≈ [81.89, 92.29].", "meta": {"id": "WUA_STAT_25K_014274", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=10.09, s=18.58, n=191, compute CI."}]}, "response": "95% CI ≈ [7.45, 12.73].", "meta": {"id": "WUA_STAT_25K_014275", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.096, sens=0.71, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.485.", "meta": {"id": "WUA_STAT_25K_014276", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.148, sens=0.73, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.427.", "meta": {"id": "WUA_STAT_25K_014277", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.58, compute E and Var."}]}, "response": "E[X]=22.04; Var[X]=9.257.", "meta": {"id": "WUA_STAT_25K_014278", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.38, μ=22.24, σ=22.59, compute z."}]}, "response": "z = 3.326.", "meta": {"id": "WUA_STAT_25K_014279", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.63, compute E and Var."}]}, "response": "E[X]=23.94; Var[X]=8.858.", "meta": {"id": "WUA_STAT_25K_014280", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.69, compute E and Var."}]}, "response": "E[X]=26.91; Var[X]=8.342.", "meta": {"id": "WUA_STAT_25K_014281", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.78, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.241.", "meta": {"id": "WUA_STAT_25K_014282", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.18, sens=0.86, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.611.", "meta": {"id": "WUA_STAT_25K_014283", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.52, μ=21.04, σ=1.44, compute z."}]}, "response": "z = 37.139.", "meta": {"id": "WUA_STAT_25K_014284", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.94, s=39.81, n=44, compute CI."}]}, "response": "95% CI ≈ [30.18, 53.70].", "meta": {"id": "WUA_STAT_25K_014285", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.95, μ=28.06, σ=13.49, compute z."}]}, "response": "z = 2.512.", "meta": {"id": "WUA_STAT_25K_014286", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.77, μ=21.44, σ=3.67, compute z."}]}, "response": "z = 15.076.", "meta": {"id": "WUA_STAT_25K_014287", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.12, μ=87.71, σ=21.12, compute z."}]}, "response": "z = -0.170.", "meta": {"id": "WUA_STAT_25K_014288", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.95, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.315.", "meta": {"id": "WUA_STAT_25K_014289", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.29, μ=72.47, σ=11.27, compute z."}]}, "response": "z = 0.871.", "meta": {"id": "WUA_STAT_25K_014290", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.016, sens=0.88, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.054.", "meta": {"id": "WUA_STAT_25K_014291", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.89, μ=67.6, σ=2.75, compute z."}]}, "response": "z = -16.258.", "meta": {"id": "WUA_STAT_25K_014292", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.76, s=28.36, n=169, compute CI."}]}, "response": "95% CI ≈ [169.48, 178.04].", "meta": {"id": "WUA_STAT_25K_014293", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.07, μ=94.66, σ=17.49, compute z."}]}, "response": "z = -2.664.", "meta": {"id": "WUA_STAT_25K_014294", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.93, μ=67.56, σ=5.36, compute z."}]}, "response": "z = 4.920.", "meta": {"id": "WUA_STAT_25K_014295", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.248, sens=0.93, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.514.", "meta": {"id": "WUA_STAT_25K_014296", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.71, compute E and Var."}]}, "response": "E[X]=7.10; Var[X]=2.059.", "meta": {"id": "WUA_STAT_25K_014297", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.171, sens=0.95, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.797.", "meta": {"id": "WUA_STAT_25K_014298", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.61, μ=37.73, σ=24.24, compute z."}]}, "response": "z = -0.005.", "meta": {"id": "WUA_STAT_25K_014299", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.64, s=34.64, n=21, compute CI."}]}, "response": "95% CI ≈ [166.82, 196.46].", "meta": {"id": "WUA_STAT_25K_014300", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.63, s=10.68, n=17, compute CI."}]}, "response": "95% CI ≈ [48.55, 58.71].", "meta": {"id": "WUA_STAT_25K_014301", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.02, sens=0.86, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.099.", "meta": {"id": "WUA_STAT_25K_014302", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.057, sens=0.8, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.162.", "meta": {"id": "WUA_STAT_25K_014303", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.37, compute E and Var."}]}, "response": "E[X]=13.69; Var[X]=8.625.", "meta": {"id": "WUA_STAT_25K_014304", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.23, μ=85.38, σ=12.04, compute z."}]}, "response": "z = -3.999.", "meta": {"id": "WUA_STAT_25K_014305", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.38, μ=23.73, σ=11.36, compute z."}]}, "response": "z = -0.823.", "meta": {"id": "WUA_STAT_25K_014306", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.68, compute E and Var."}]}, "response": "E[X]=25.84; Var[X]=8.269.", "meta": {"id": "WUA_STAT_25K_014307", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.11, compute E and Var."}]}, "response": "E[X]=2.20; Var[X]=1.958.", "meta": {"id": "WUA_STAT_25K_014308", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.217, sens=0.88, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.515.", "meta": {"id": "WUA_STAT_25K_014309", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.085, sens=0.89, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.222.", "meta": {"id": "WUA_STAT_25K_014310", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=103.1, s=19.77, n=94, compute CI."}]}, "response": "95% CI ≈ [99.10, 107.10].", "meta": {"id": "WUA_STAT_25K_014311", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.29, μ=73.66, σ=22.03, compute z."}]}, "response": "z = -0.562.", "meta": {"id": "WUA_STAT_25K_014312", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.68, μ=97.03, σ=15.4, compute z."}]}, "response": "z = -5.477.", "meta": {"id": "WUA_STAT_25K_014313", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.015, sens=0.76, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.095.", "meta": {"id": "WUA_STAT_25K_014314", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.49, compute E and Var."}]}, "response": "E[X]=16.17; Var[X]=8.247.", "meta": {"id": "WUA_STAT_25K_014315", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.07, sens=0.72, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.191.", "meta": {"id": "WUA_STAT_25K_014316", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.08, compute E and Var."}]}, "response": "E[X]=0.80; Var[X]=0.736.", "meta": {"id": "WUA_STAT_25K_014317", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.45, μ=97.7, σ=18.92, compute z."}]}, "response": "z = -0.383.", "meta": {"id": "WUA_STAT_25K_014318", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.85, μ=14.33, σ=4.37, compute z."}]}, "response": "z = -1.712.", "meta": {"id": "WUA_STAT_25K_014319", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.221, sens=0.93, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.557.", "meta": {"id": "WUA_STAT_25K_014320", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=10.25, s=36.07, n=189, compute CI."}]}, "response": "95% CI ≈ [5.11, 15.39].", "meta": {"id": "WUA_STAT_25K_014321", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.64, s=6.0, n=140, compute CI."}]}, "response": "95% CI ≈ [36.65, 38.63].", "meta": {"id": "WUA_STAT_25K_014322", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.38, compute E and Var."}]}, "response": "E[X]=3.42; Var[X]=2.120.", "meta": {"id": "WUA_STAT_25K_014323", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.92, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.551.", "meta": {"id": "WUA_STAT_25K_014324", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.97, s=3.3, n=95, compute CI."}]}, "response": "95% CI ≈ [106.31, 107.63].", "meta": {"id": "WUA_STAT_25K_014325", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.23, compute E and Var."}]}, "response": "E[X]=2.76; Var[X]=2.125.", "meta": {"id": "WUA_STAT_25K_014326", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=72.0, s=19.2, n=28, compute CI."}]}, "response": "95% CI ≈ [64.89, 79.11].", "meta": {"id": "WUA_STAT_25K_014327", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.179, sens=0.81, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.855.", "meta": {"id": "WUA_STAT_25K_014328", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=196.29, s=18.03, n=46, compute CI."}]}, "response": "95% CI ≈ [191.08, 201.50].", "meta": {"id": "WUA_STAT_25K_014329", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.33, compute E and Var."}]}, "response": "E[X]=11.88; Var[X]=7.960.", "meta": {"id": "WUA_STAT_25K_014330", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.83, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.076.", "meta": {"id": "WUA_STAT_25K_014331", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.69, s=34.67, n=97, compute CI."}]}, "response": "95% CI ≈ [149.79, 163.59].", "meta": {"id": "WUA_STAT_25K_014332", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.59, s=21.44, n=131, compute CI."}]}, "response": "95% CI ≈ [172.92, 180.26].", "meta": {"id": "WUA_STAT_25K_014333", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.11, μ=7.3, σ=20.64, compute z."}]}, "response": "z = 2.849.", "meta": {"id": "WUA_STAT_25K_014334", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.05, compute E and Var."}]}, "response": "E[X]=1.95; Var[X]=1.853.", "meta": {"id": "WUA_STAT_25K_014335", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.37, compute E and Var."}]}, "response": "E[X]=6.66; Var[X]=4.196.", "meta": {"id": "WUA_STAT_25K_014336", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.185, sens=0.95, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.606.", "meta": {"id": "WUA_STAT_25K_014337", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.22, s=20.91, n=41, compute CI."}]}, "response": "95% CI ≈ [27.82, 40.62].", "meta": {"id": "WUA_STAT_25K_014338", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.08, compute E and Var."}]}, "response": "E[X]=2.72; Var[X]=2.502.", "meta": {"id": "WUA_STAT_25K_014339", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.24, s=29.58, n=31, compute CI."}]}, "response": "95% CI ≈ [166.83, 187.65].", "meta": {"id": "WUA_STAT_25K_014340", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.77, compute E and Var."}]}, "response": "E[X]=18.48; Var[X]=4.250.", "meta": {"id": "WUA_STAT_25K_014341", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.94, s=30.56, n=152, compute CI."}]}, "response": "95% CI ≈ [138.08, 147.80].", "meta": {"id": "WUA_STAT_25K_014342", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.7, compute E and Var."}]}, "response": "E[X]=21.00; Var[X]=6.300.", "meta": {"id": "WUA_STAT_25K_014343", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.61, compute E and Var."}]}, "response": "E[X]=23.18; Var[X]=9.040.", "meta": {"id": "WUA_STAT_25K_014344", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.48, compute E and Var."}]}, "response": "E[X]=3.36; Var[X]=1.747.", "meta": {"id": "WUA_STAT_25K_014345", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.72, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.802.", "meta": {"id": "WUA_STAT_25K_014346", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=141.6, s=16.67, n=52, compute CI."}]}, "response": "95% CI ≈ [137.07, 146.13].", "meta": {"id": "WUA_STAT_25K_014347", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.95, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.390.", "meta": {"id": "WUA_STAT_25K_014348", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.123, sens=0.94, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.594.", "meta": {"id": "WUA_STAT_25K_014349", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.67, s=30.27, n=80, compute CI."}]}, "response": "95% CI ≈ [55.04, 68.30].", "meta": {"id": "WUA_STAT_25K_014350", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.38, s=13.95, n=188, compute CI."}]}, "response": "95% CI ≈ [88.39, 92.37].", "meta": {"id": "WUA_STAT_25K_014351", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.89, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.440.", "meta": {"id": "WUA_STAT_25K_014352", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.72, s=1.52, n=192, compute CI."}]}, "response": "95% CI ≈ [188.50, 188.94].", "meta": {"id": "WUA_STAT_25K_014353", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.24, compute E and Var."}]}, "response": "E[X]=5.28; Var[X]=4.013.", "meta": {"id": "WUA_STAT_25K_014354", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.91, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.243.", "meta": {"id": "WUA_STAT_25K_014355", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.26, compute E and Var."}]}, "response": "E[X]=2.34; Var[X]=1.732.", "meta": {"id": "WUA_STAT_25K_014356", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.95, μ=4.96, σ=24.4, compute z."}]}, "response": "z = 1.926.", "meta": {"id": "WUA_STAT_25K_014357", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.04, μ=95.75, σ=15.87, compute z."}]}, "response": "z = -0.108.", "meta": {"id": "WUA_STAT_25K_014358", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.44, compute E and Var."}]}, "response": "E[X]=14.96; Var[X]=8.378.", "meta": {"id": "WUA_STAT_25K_014359", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.78, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.686.", "meta": {"id": "WUA_STAT_25K_014360", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.029, sens=0.94, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.412.", "meta": {"id": "WUA_STAT_25K_014361", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.96, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.777.", "meta": {"id": "WUA_STAT_25K_014362", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.27, compute E and Var."}]}, "response": "E[X]=3.78; Var[X]=2.759.", "meta": {"id": "WUA_STAT_25K_014363", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.13, compute E and Var."}]}, "response": "E[X]=3.64; Var[X]=3.167.", "meta": {"id": "WUA_STAT_25K_014364", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.39, compute E and Var."}]}, "response": "E[X]=3.90; Var[X]=2.379.", "meta": {"id": "WUA_STAT_25K_014365", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.06, compute E and Var."}]}, "response": "E[X]=0.48; Var[X]=0.451.", "meta": {"id": "WUA_STAT_25K_014366", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=64.02, s=28.06, n=138, compute CI."}]}, "response": "95% CI ≈ [59.34, 68.70].", "meta": {"id": "WUA_STAT_25K_014367", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.32, compute E and Var."}]}, "response": "E[X]=11.20; Var[X]=7.616.", "meta": {"id": "WUA_STAT_25K_014368", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.19, compute E and Var."}]}, "response": "E[X]=3.42; Var[X]=2.770.", "meta": {"id": "WUA_STAT_25K_014369", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.137, sens=0.77, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.289.", "meta": {"id": "WUA_STAT_25K_014370", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.36, compute E and Var."}]}, "response": "E[X]=3.24; Var[X]=2.074.", "meta": {"id": "WUA_STAT_25K_014371", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.19, compute E and Var."}]}, "response": "E[X]=1.14; Var[X]=0.923.", "meta": {"id": "WUA_STAT_25K_014372", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=100.0, μ=81.82, σ=19.47, compute z."}]}, "response": "z = 0.934.", "meta": {"id": "WUA_STAT_25K_014373", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.195, sens=0.93, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.518.", "meta": {"id": "WUA_STAT_25K_014374", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.41, compute E and Var."}]}, "response": "E[X]=12.30; Var[X]=7.257.", "meta": {"id": "WUA_STAT_25K_014375", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.45, compute E and Var."}]}, "response": "E[X]=5.85; Var[X]=3.218.", "meta": {"id": "WUA_STAT_25K_014376", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.91, s=7.23, n=88, compute CI."}]}, "response": "95% CI ≈ [34.40, 37.42].", "meta": {"id": "WUA_STAT_25K_014377", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=71.73, s=14.87, n=106, compute CI."}]}, "response": "95% CI ≈ [68.90, 74.56].", "meta": {"id": "WUA_STAT_25K_014378", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.134, sens=0.8, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.421.", "meta": {"id": "WUA_STAT_25K_014379", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.245, sens=0.97, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.677.", "meta": {"id": "WUA_STAT_25K_014380", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.0, s=15.61, n=187, compute CI."}]}, "response": "95% CI ≈ [33.76, 38.24].", "meta": {"id": "WUA_STAT_25K_014381", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.11, compute E and Var."}]}, "response": "E[X]=3.08; Var[X]=2.741.", "meta": {"id": "WUA_STAT_25K_014382", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.24, μ=26.21, σ=7.81, compute z."}]}, "response": "z = -0.764.", "meta": {"id": "WUA_STAT_25K_014383", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.22, s=16.41, n=27, compute CI."}]}, "response": "95% CI ≈ [79.03, 91.41].", "meta": {"id": "WUA_STAT_25K_014384", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.44, compute E and Var."}]}, "response": "E[X]=15.40; Var[X]=8.624.", "meta": {"id": "WUA_STAT_25K_014385", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.18, sens=0.84, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.672.", "meta": {"id": "WUA_STAT_25K_014386", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.35, μ=17.83, σ=3.53, compute z."}]}, "response": "z = 0.714.", "meta": {"id": "WUA_STAT_25K_014387", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.07, s=34.55, n=120, compute CI."}]}, "response": "95% CI ≈ [186.89, 199.25].", "meta": {"id": "WUA_STAT_25K_014388", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.77, μ=91.18, σ=20.13, compute z."}]}, "response": "z = -4.193.", "meta": {"id": "WUA_STAT_25K_014389", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=115.23, s=35.92, n=155, compute CI."}]}, "response": "95% CI ≈ [109.58, 120.88].", "meta": {"id": "WUA_STAT_25K_014390", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.55, compute E and Var."}]}, "response": "E[X]=15.40; Var[X]=6.930.", "meta": {"id": "WUA_STAT_25K_014391", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.86, μ=96.84, σ=21.79, compute z."}]}, "response": "z = -1.697.", "meta": {"id": "WUA_STAT_25K_014392", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.94, μ=55.13, σ=3.79, compute z."}]}, "response": "z = -8.493.", "meta": {"id": "WUA_STAT_25K_014393", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.014, sens=0.88, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.122.", "meta": {"id": "WUA_STAT_25K_014394", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.7, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.553.", "meta": {"id": "WUA_STAT_25K_014395", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.99, μ=89.58, σ=9.02, compute z."}]}, "response": "z = -0.065.", "meta": {"id": "WUA_STAT_25K_014396", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.22, s=22.54, n=130, compute CI."}]}, "response": "95% CI ≈ [190.35, 198.09].", "meta": {"id": "WUA_STAT_25K_014397", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.36, μ=57.98, σ=12.77, compute z."}]}, "response": "z = -3.181.", "meta": {"id": "WUA_STAT_25K_014398", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.44, μ=40.11, σ=1.47, compute z."}]}, "response": "z = -16.782.", "meta": {"id": "WUA_STAT_25K_014399", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.087, sens=0.93, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.318.", "meta": {"id": "WUA_STAT_25K_014400", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.38, compute E and Var."}]}, "response": "E[X]=9.50; Var[X]=5.890.", "meta": {"id": "WUA_STAT_25K_014401", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.72, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.033.", "meta": {"id": "WUA_STAT_25K_014402", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.05, compute E and Var."}]}, "response": "E[X]=1.70; Var[X]=1.615.", "meta": {"id": "WUA_STAT_25K_014403", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.234, sens=0.92, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.738.", "meta": {"id": "WUA_STAT_25K_014404", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.89, s=12.71, n=99, compute CI."}]}, "response": "95% CI ≈ [186.39, 191.39].", "meta": {"id": "WUA_STAT_25K_014405", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.63, μ=64.59, σ=5.49, compute z."}]}, "response": "z = -3.454.", "meta": {"id": "WUA_STAT_25K_014406", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.79, s=17.89, n=76, compute CI."}]}, "response": "95% CI ≈ [194.77, 202.81].", "meta": {"id": "WUA_STAT_25K_014407", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.31, s=37.61, n=30, compute CI."}]}, "response": "95% CI ≈ [105.85, 132.77].", "meta": {"id": "WUA_STAT_25K_014408", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.17, compute E and Var."}]}, "response": "E[X]=5.78; Var[X]=4.797.", "meta": {"id": "WUA_STAT_25K_014409", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=158.22, s=10.57, n=90, compute CI."}]}, "response": "95% CI ≈ [156.04, 160.40].", "meta": {"id": "WUA_STAT_25K_014410", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.71, compute E and Var."}]}, "response": "E[X]=25.56; Var[X]=7.412.", "meta": {"id": "WUA_STAT_25K_014411", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.69, compute E and Var."}]}, "response": "E[X]=18.63; Var[X]=5.775.", "meta": {"id": "WUA_STAT_25K_014412", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.61, μ=63.79, σ=19.15, compute z."}]}, "response": "z = -3.143.", "meta": {"id": "WUA_STAT_25K_014413", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.051, sens=0.8, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.152.", "meta": {"id": "WUA_STAT_25K_014414", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.5, μ=17.65, σ=24.03, compute z."}]}, "response": "z = 1.866.", "meta": {"id": "WUA_STAT_25K_014415", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.12, s=10.43, n=19, compute CI."}]}, "response": "95% CI ≈ [82.43, 91.81].", "meta": {"id": "WUA_STAT_25K_014416", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.016, sens=0.9, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.084.", "meta": {"id": "WUA_STAT_25K_014417", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.76, compute E and Var."}]}, "response": "E[X]=17.48; Var[X]=4.195.", "meta": {"id": "WUA_STAT_25K_014418", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.3, compute E and Var."}]}, "response": "E[X]=9.90; Var[X]=6.930.", "meta": {"id": "WUA_STAT_25K_014419", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.21, compute E and Var."}]}, "response": "E[X]=7.35; Var[X]=5.806.", "meta": {"id": "WUA_STAT_25K_014420", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.156, sens=0.76, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.379.", "meta": {"id": "WUA_STAT_25K_014421", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.73, s=12.5, n=24, compute CI."}]}, "response": "95% CI ≈ [140.73, 150.73].", "meta": {"id": "WUA_STAT_25K_014422", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.7, compute E and Var."}]}, "response": "E[X]=15.40; Var[X]=4.620.", "meta": {"id": "WUA_STAT_25K_014423", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.65, μ=60.66, σ=9.78, compute z."}]}, "response": "z = 2.555.", "meta": {"id": "WUA_STAT_25K_014424", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.55, compute E and Var."}]}, "response": "E[X]=7.70; Var[X]=3.465.", "meta": {"id": "WUA_STAT_25K_014425", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.6, compute E and Var."}]}, "response": "E[X]=9.00; Var[X]=3.600.", "meta": {"id": "WUA_STAT_25K_014426", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.19, compute E and Var."}]}, "response": "E[X]=1.71; Var[X]=1.385.", "meta": {"id": "WUA_STAT_25K_014427", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.158, sens=0.7, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.451.", "meta": {"id": "WUA_STAT_25K_014428", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.71, μ=6.12, σ=13.15, compute z."}]}, "response": "z = 4.760.", "meta": {"id": "WUA_STAT_25K_014429", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.27, s=36.8, n=144, compute CI."}]}, "response": "95% CI ≈ [123.26, 135.28].", "meta": {"id": "WUA_STAT_25K_014430", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.57, s=25.26, n=78, compute CI."}]}, "response": "95% CI ≈ [188.96, 200.18].", "meta": {"id": "WUA_STAT_25K_014431", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.2, s=6.64, n=50, compute CI."}]}, "response": "95% CI ≈ [137.36, 141.04].", "meta": {"id": "WUA_STAT_25K_014432", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.99, s=38.19, n=29, compute CI."}]}, "response": "95% CI ≈ [28.09, 55.89].", "meta": {"id": "WUA_STAT_25K_014433", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.73, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.314.", "meta": {"id": "WUA_STAT_25K_014434", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=144.32, s=30.59, n=126, compute CI."}]}, "response": "95% CI ≈ [138.98, 149.66].", "meta": {"id": "WUA_STAT_25K_014435", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.62, μ=10.44, σ=7.41, compute z."}]}, "response": "z = 9.336.", "meta": {"id": "WUA_STAT_25K_014436", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.123, sens=0.95, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.816.", "meta": {"id": "WUA_STAT_25K_014437", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.28, μ=81.19, σ=23.76, compute z."}]}, "response": "z = -0.375.", "meta": {"id": "WUA_STAT_25K_014438", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=124.89, s=29.51, n=194, compute CI."}]}, "response": "95% CI ≈ [120.74, 129.04].", "meta": {"id": "WUA_STAT_25K_014439", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.092, sens=0.83, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.375.", "meta": {"id": "WUA_STAT_25K_014440", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=26.39, s=29.97, n=128, compute CI."}]}, "response": "95% CI ≈ [21.20, 31.58].", "meta": {"id": "WUA_STAT_25K_014441", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.44, compute E and Var."}]}, "response": "E[X]=17.60; Var[X]=9.856.", "meta": {"id": "WUA_STAT_25K_014442", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.42, compute E and Var."}]}, "response": "E[X]=11.34; Var[X]=6.577.", "meta": {"id": "WUA_STAT_25K_014443", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.27, s=7.6, n=166, compute CI."}]}, "response": "95% CI ≈ [149.11, 151.43].", "meta": {"id": "WUA_STAT_25K_014444", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.058, sens=0.77, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.177.", "meta": {"id": "WUA_STAT_25K_014445", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.81, μ=96.52, σ=22.23, compute z."}]}, "response": "z = -2.956.", "meta": {"id": "WUA_STAT_25K_014446", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.37, μ=54.04, σ=11.61, compute z."}]}, "response": "z = 3.646.", "meta": {"id": "WUA_STAT_25K_014447", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.88, s=5.39, n=116, compute CI."}]}, "response": "95% CI ≈ [187.90, 189.86].", "meta": {"id": "WUA_STAT_25K_014448", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.38, μ=17.24, σ=3.57, compute z."}]}, "response": "z = 9.843.", "meta": {"id": "WUA_STAT_25K_014449", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.78, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.629.", "meta": {"id": "WUA_STAT_25K_014450", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.72, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.746.", "meta": {"id": "WUA_STAT_25K_014451", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.94, s=14.49, n=52, compute CI."}]}, "response": "95% CI ≈ [41.00, 48.88].", "meta": {"id": "WUA_STAT_25K_014452", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.17, s=4.92, n=148, compute CI."}]}, "response": "95% CI ≈ [111.38, 112.96].", "meta": {"id": "WUA_STAT_25K_014453", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.44, compute E and Var."}]}, "response": "E[X]=8.36; Var[X]=4.682.", "meta": {"id": "WUA_STAT_25K_014454", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=195.28, s=36.19, n=160, compute CI."}]}, "response": "95% CI ≈ [189.67, 200.89].", "meta": {"id": "WUA_STAT_25K_014455", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.21, μ=84.67, σ=3.31, compute z."}]}, "response": "z = -23.704.", "meta": {"id": "WUA_STAT_25K_014456", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.13, compute E and Var."}]}, "response": "E[X]=3.64; Var[X]=3.167.", "meta": {"id": "WUA_STAT_25K_014457", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.43, compute E and Var."}]}, "response": "E[X]=9.46; Var[X]=5.392.", "meta": {"id": "WUA_STAT_25K_014458", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.16, μ=6.91, σ=2.32, compute z."}]}, "response": "z = 10.884.", "meta": {"id": "WUA_STAT_25K_014459", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.86, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.586.", "meta": {"id": "WUA_STAT_25K_014460", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.12, compute E and Var."}]}, "response": "E[X]=1.44; Var[X]=1.267.", "meta": {"id": "WUA_STAT_25K_014461", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.66, s=37.89, n=136, compute CI."}]}, "response": "95% CI ≈ [56.29, 69.03].", "meta": {"id": "WUA_STAT_25K_014462", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.51, μ=44.97, σ=2.66, compute z."}]}, "response": "z = 1.331.", "meta": {"id": "WUA_STAT_25K_014463", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.18, μ=70.8, σ=1.93, compute z."}]}, "response": "z = -13.793.", "meta": {"id": "WUA_STAT_25K_014464", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.29, s=7.52, n=87, compute CI."}]}, "response": "95% CI ≈ [28.71, 31.87].", "meta": {"id": "WUA_STAT_25K_014465", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.84, μ=71.31, σ=17.87, compute z."}]}, "response": "z = 0.309.", "meta": {"id": "WUA_STAT_25K_014466", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.14, compute E and Var."}]}, "response": "E[X]=3.36; Var[X]=2.890.", "meta": {"id": "WUA_STAT_25K_014467", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.19, μ=73.47, σ=18.8, compute z."}]}, "response": "z = 0.783.", "meta": {"id": "WUA_STAT_25K_014468", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.49, μ=10.34, σ=13.81, compute z."}]}, "response": "z = 6.455.", "meta": {"id": "WUA_STAT_25K_014469", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.1, μ=4.85, σ=1.22, compute z."}]}, "response": "z = 10.861.", "meta": {"id": "WUA_STAT_25K_014470", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.93, μ=82.45, σ=21.94, compute z."}]}, "response": "z = -2.941.", "meta": {"id": "WUA_STAT_25K_014471", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.74, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.266.", "meta": {"id": "WUA_STAT_25K_014472", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.87, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.551.", "meta": {"id": "WUA_STAT_25K_014473", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.048, sens=0.78, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.663.", "meta": {"id": "WUA_STAT_25K_014474", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.14, s=8.39, n=51, compute CI."}]}, "response": "95% CI ≈ [190.84, 195.44].", "meta": {"id": "WUA_STAT_25K_014475", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.31, compute E and Var."}]}, "response": "E[X]=8.68; Var[X]=5.989.", "meta": {"id": "WUA_STAT_25K_014476", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.44, compute E and Var."}]}, "response": "E[X]=12.32; Var[X]=6.899.", "meta": {"id": "WUA_STAT_25K_014477", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.034, sens=0.8, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.484.", "meta": {"id": "WUA_STAT_25K_014478", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.67, μ=82.32, σ=14.51, compute z."}]}, "response": "z = -4.938.", "meta": {"id": "WUA_STAT_25K_014479", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.07, compute E and Var."}]}, "response": "E[X]=1.19; Var[X]=1.107.", "meta": {"id": "WUA_STAT_25K_014480", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.232, sens=0.84, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.572.", "meta": {"id": "WUA_STAT_25K_014481", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.09, s=19.82, n=125, compute CI."}]}, "response": "95% CI ≈ [173.62, 180.56].", "meta": {"id": "WUA_STAT_25K_014482", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.81, μ=9.39, σ=12.57, compute z."}]}, "response": "z = 2.500.", "meta": {"id": "WUA_STAT_25K_014483", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.38, compute E and Var."}]}, "response": "E[X]=10.64; Var[X]=6.597.", "meta": {"id": "WUA_STAT_25K_014484", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.171, sens=0.84, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.374.", "meta": {"id": "WUA_STAT_25K_014485", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.2, compute E and Var."}]}, "response": "E[X]=5.40; Var[X]=4.320.", "meta": {"id": "WUA_STAT_25K_014486", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=103.65, s=39.49, n=35, compute CI."}]}, "response": "95% CI ≈ [90.57, 116.73].", "meta": {"id": "WUA_STAT_25K_014487", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.2, compute E and Var."}]}, "response": "E[X]=4.20; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_014488", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.021, sens=0.77, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.067.", "meta": {"id": "WUA_STAT_25K_014489", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.123, sens=0.82, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.469.", "meta": {"id": "WUA_STAT_25K_014490", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.066, sens=0.99, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.350.", "meta": {"id": "WUA_STAT_25K_014491", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.238, sens=0.81, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.484.", "meta": {"id": "WUA_STAT_25K_014492", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.75, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.125.", "meta": {"id": "WUA_STAT_25K_014493", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.33, s=4.41, n=13, compute CI."}]}, "response": "95% CI ≈ [98.93, 103.73].", "meta": {"id": "WUA_STAT_25K_014494", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.77, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.514.", "meta": {"id": "WUA_STAT_25K_014495", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.036, sens=0.97, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.475.", "meta": {"id": "WUA_STAT_25K_014496", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.78, μ=11.05, σ=19.4, compute z."}]}, "response": "z = 2.460.", "meta": {"id": "WUA_STAT_25K_014497", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=158.41, s=26.16, n=157, compute CI."}]}, "response": "95% CI ≈ [154.32, 162.50].", "meta": {"id": "WUA_STAT_25K_014498", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.13, s=6.93, n=77, compute CI."}]}, "response": "95% CI ≈ [171.58, 174.68].", "meta": {"id": "WUA_STAT_25K_014499", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.29, μ=85.92, σ=3.51, compute z."}]}, "response": "z = -14.994.", "meta": {"id": "WUA_STAT_25K_014500", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.65, compute E and Var."}]}, "response": "E[X]=26.00; Var[X]=9.100.", "meta": {"id": "WUA_STAT_25K_014501", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.44, μ=74.34, σ=10.69, compute z."}]}, "response": "z = 2.067.", "meta": {"id": "WUA_STAT_25K_014502", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.98, μ=3.7, σ=8.32, compute z."}]}, "response": "z = 4.601.", "meta": {"id": "WUA_STAT_25K_014503", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.26, s=3.31, n=144, compute CI."}]}, "response": "95% CI ≈ [133.72, 134.80].", "meta": {"id": "WUA_STAT_25K_014504", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.85, μ=60.11, σ=13.6, compute z."}]}, "response": "z = -2.593.", "meta": {"id": "WUA_STAT_25K_014505", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.65, compute E and Var."}]}, "response": "E[X]=11.70; Var[X]=4.095.", "meta": {"id": "WUA_STAT_25K_014506", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.79, μ=57.34, σ=15.3, compute z."}]}, "response": "z = -3.042.", "meta": {"id": "WUA_STAT_25K_014507", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.33, μ=75.87, σ=12.21, compute z."}]}, "response": "z = -2.010.", "meta": {"id": "WUA_STAT_25K_014508", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.77, compute E and Var."}]}, "response": "E[X]=19.25; Var[X]=4.427.", "meta": {"id": "WUA_STAT_25K_014509", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.17, compute E and Var."}]}, "response": "E[X]=1.02; Var[X]=0.847.", "meta": {"id": "WUA_STAT_25K_014510", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=147.26, s=12.73, n=19, compute CI."}]}, "response": "95% CI ≈ [141.54, 152.98].", "meta": {"id": "WUA_STAT_25K_014511", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.56, s=8.78, n=178, compute CI."}]}, "response": "95% CI ≈ [17.27, 19.85].", "meta": {"id": "WUA_STAT_25K_014512", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.76, μ=82.61, σ=22.09, compute z."}]}, "response": "z = -0.446.", "meta": {"id": "WUA_STAT_25K_014513", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.223, sens=0.81, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.482.", "meta": {"id": "WUA_STAT_25K_014514", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.42, compute E and Var."}]}, "response": "E[X]=3.36; Var[X]=1.949.", "meta": {"id": "WUA_STAT_25K_014515", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.95, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.416.", "meta": {"id": "WUA_STAT_25K_014516", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.52, compute E and Var."}]}, "response": "E[X]=3.12; Var[X]=1.498.", "meta": {"id": "WUA_STAT_25K_014517", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.1, compute E and Var."}]}, "response": "E[X]=3.20; Var[X]=2.880.", "meta": {"id": "WUA_STAT_25K_014518", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.1, μ=93.67, σ=23.39, compute z."}]}, "response": "z = -1.692.", "meta": {"id": "WUA_STAT_25K_014519", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.76, s=20.74, n=96, compute CI."}]}, "response": "95% CI ≈ [166.61, 174.91].", "meta": {"id": "WUA_STAT_25K_014520", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.81, s=23.05, n=35, compute CI."}]}, "response": "95% CI ≈ [118.17, 133.45].", "meta": {"id": "WUA_STAT_25K_014521", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.95, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.674.", "meta": {"id": "WUA_STAT_25K_014522", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.83, s=2.32, n=22, compute CI."}]}, "response": "95% CI ≈ [22.86, 24.80].", "meta": {"id": "WUA_STAT_25K_014523", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.71, compute E and Var."}]}, "response": "E[X]=26.27; Var[X]=7.618.", "meta": {"id": "WUA_STAT_25K_014524", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=100.98, s=13.06, n=17, compute CI."}]}, "response": "95% CI ≈ [94.77, 107.19].", "meta": {"id": "WUA_STAT_25K_014525", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.74, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.375.", "meta": {"id": "WUA_STAT_25K_014526", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.49, compute E and Var."}]}, "response": "E[X]=7.84; Var[X]=3.998.", "meta": {"id": "WUA_STAT_25K_014527", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.41, compute E and Var."}]}, "response": "E[X]=11.07; Var[X]=6.531.", "meta": {"id": "WUA_STAT_25K_014528", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.237, sens=0.98, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.603.", "meta": {"id": "WUA_STAT_25K_014529", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.76, s=9.37, n=53, compute CI."}]}, "response": "95% CI ≈ [12.24, 17.28].", "meta": {"id": "WUA_STAT_25K_014530", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.34, μ=64.71, σ=4.98, compute z."}]}, "response": "z = -0.275.", "meta": {"id": "WUA_STAT_25K_014531", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.09, μ=43.9, σ=9.72, compute z."}]}, "response": "z = -0.906.", "meta": {"id": "WUA_STAT_25K_014532", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.86, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.325.", "meta": {"id": "WUA_STAT_25K_014533", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.91, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.371.", "meta": {"id": "WUA_STAT_25K_014534", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.03, sens=0.94, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.122.", "meta": {"id": "WUA_STAT_25K_014535", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.28, s=36.46, n=125, compute CI."}]}, "response": "95% CI ≈ [153.89, 166.67].", "meta": {"id": "WUA_STAT_25K_014536", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.031, sens=0.79, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.092.", "meta": {"id": "WUA_STAT_25K_014537", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.68, μ=65.78, σ=7.01, compute z."}]}, "response": "z = 1.555.", "meta": {"id": "WUA_STAT_25K_014538", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.087, sens=0.84, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.258.", "meta": {"id": "WUA_STAT_25K_014539", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.18, μ=60.49, σ=21.81, compute z."}]}, "response": "z = -0.656.", "meta": {"id": "WUA_STAT_25K_014540", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.35, s=12.87, n=181, compute CI."}]}, "response": "95% CI ≈ [34.48, 38.22].", "meta": {"id": "WUA_STAT_25K_014541", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.39, μ=59.14, σ=12.09, compute z."}]}, "response": "z = -1.964.", "meta": {"id": "WUA_STAT_25K_014542", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.089, sens=0.78, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.718.", "meta": {"id": "WUA_STAT_25K_014543", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.55, μ=27.31, σ=11.06, compute z."}]}, "response": "z = 2.373.", "meta": {"id": "WUA_STAT_25K_014544", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.77, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.476.", "meta": {"id": "WUA_STAT_25K_014545", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.66, compute E and Var."}]}, "response": "E[X]=25.74; Var[X]=8.752.", "meta": {"id": "WUA_STAT_25K_014546", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=147.71, s=37.95, n=131, compute CI."}]}, "response": "95% CI ≈ [141.21, 154.21].", "meta": {"id": "WUA_STAT_25K_014547", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.222, sens=0.98, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.778.", "meta": {"id": "WUA_STAT_25K_014548", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.24, sens=0.97, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.793.", "meta": {"id": "WUA_STAT_25K_014549", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.159, sens=0.92, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.453.", "meta": {"id": "WUA_STAT_25K_014550", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=165.53, s=30.62, n=112, compute CI."}]}, "response": "95% CI ≈ [159.86, 171.20].", "meta": {"id": "WUA_STAT_25K_014551", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.22, compute E and Var."}]}, "response": "E[X]=5.28; Var[X]=4.118.", "meta": {"id": "WUA_STAT_25K_014552", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.54, compute E and Var."}]}, "response": "E[X]=21.06; Var[X]=9.688.", "meta": {"id": "WUA_STAT_25K_014553", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.027, sens=0.71, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.062.", "meta": {"id": "WUA_STAT_25K_014554", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.11, compute E and Var."}]}, "response": "E[X]=1.43; Var[X]=1.273.", "meta": {"id": "WUA_STAT_25K_014555", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.03, μ=48.5, σ=16.25, compute z."}]}, "response": "z = 1.940.", "meta": {"id": "WUA_STAT_25K_014556", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.064, sens=0.78, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.182.", "meta": {"id": "WUA_STAT_25K_014557", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.67, μ=32.21, σ=17.56, compute z."}]}, "response": "z = -0.942.", "meta": {"id": "WUA_STAT_25K_014558", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.82, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.319.", "meta": {"id": "WUA_STAT_25K_014559", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.83, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.478.", "meta": {"id": "WUA_STAT_25K_014560", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.92, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.427.", "meta": {"id": "WUA_STAT_25K_014561", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.31, s=17.7, n=104, compute CI."}]}, "response": "95% CI ≈ [13.91, 20.71].", "meta": {"id": "WUA_STAT_25K_014562", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.54, compute E and Var."}]}, "response": "E[X]=19.44; Var[X]=8.942.", "meta": {"id": "WUA_STAT_25K_014563", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.73, compute E and Var."}]}, "response": "E[X]=22.63; Var[X]=6.110.", "meta": {"id": "WUA_STAT_25K_014564", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.087, sens=0.91, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.520.", "meta": {"id": "WUA_STAT_25K_014565", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.18, s=1.53, n=97, compute CI."}]}, "response": "95% CI ≈ [113.88, 114.48].", "meta": {"id": "WUA_STAT_25K_014566", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.41, s=16.41, n=67, compute CI."}]}, "response": "95% CI ≈ [18.48, 26.34].", "meta": {"id": "WUA_STAT_25K_014567", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=96.63, s=16.31, n=12, compute CI."}]}, "response": "95% CI ≈ [87.40, 105.86].", "meta": {"id": "WUA_STAT_25K_014568", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.44, compute E and Var."}]}, "response": "E[X]=5.28; Var[X]=2.957.", "meta": {"id": "WUA_STAT_25K_014569", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.2, s=35.02, n=119, compute CI."}]}, "response": "95% CI ≈ [169.91, 182.49].", "meta": {"id": "WUA_STAT_25K_014570", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.73, s=17.25, n=118, compute CI."}]}, "response": "95% CI ≈ [30.62, 36.84].", "meta": {"id": "WUA_STAT_25K_014571", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.016, sens=0.72, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.089.", "meta": {"id": "WUA_STAT_25K_014572", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.51, s=24.26, n=43, compute CI."}]}, "response": "95% CI ≈ [154.26, 168.76].", "meta": {"id": "WUA_STAT_25K_014573", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.16, μ=96.15, σ=22.42, compute z."}]}, "response": "z = -3.211.", "meta": {"id": "WUA_STAT_25K_014574", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.024, sens=0.92, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.201.", "meta": {"id": "WUA_STAT_25K_014575", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.76, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.174.", "meta": {"id": "WUA_STAT_25K_014576", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.04, s=1.69, n=105, compute CI."}]}, "response": "95% CI ≈ [39.72, 40.36].", "meta": {"id": "WUA_STAT_25K_014577", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.12, μ=48.51, σ=3.0, compute z."}]}, "response": "z = 16.870.", "meta": {"id": "WUA_STAT_25K_014578", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.68, compute E and Var."}]}, "response": "E[X]=6.12; Var[X]=1.958.", "meta": {"id": "WUA_STAT_25K_014579", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.119, sens=0.84, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.654.", "meta": {"id": "WUA_STAT_25K_014580", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.226, sens=0.85, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.508.", "meta": {"id": "WUA_STAT_25K_014581", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.73, s=33.58, n=11, compute CI."}]}, "response": "95% CI ≈ [64.89, 104.57].", "meta": {"id": "WUA_STAT_25K_014582", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.195, sens=0.71, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.518.", "meta": {"id": "WUA_STAT_25K_014583", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.71, μ=78.76, σ=22.56, compute z."}]}, "response": "z = -0.534.", "meta": {"id": "WUA_STAT_25K_014584", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.44, μ=27.1, σ=1.53, compute z."}]}, "response": "z = -5.007.", "meta": {"id": "WUA_STAT_25K_014585", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.019, sens=0.81, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.148.", "meta": {"id": "WUA_STAT_25K_014586", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.5, s=6.87, n=58, compute CI."}]}, "response": "95% CI ≈ [137.73, 141.27].", "meta": {"id": "WUA_STAT_25K_014587", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.28, compute E and Var."}]}, "response": "E[X]=2.80; Var[X]=2.016.", "meta": {"id": "WUA_STAT_25K_014588", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.1, compute E and Var."}]}, "response": "E[X]=2.40; Var[X]=2.160.", "meta": {"id": "WUA_STAT_25K_014589", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.84, μ=39.66, σ=10.43, compute z."}]}, "response": "z = 0.401.", "meta": {"id": "WUA_STAT_25K_014590", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.89, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.507.", "meta": {"id": "WUA_STAT_25K_014591", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.44, compute E and Var."}]}, "response": "E[X]=2.64; Var[X]=1.478.", "meta": {"id": "WUA_STAT_25K_014592", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.1, μ=66.65, σ=13.56, compute z."}]}, "response": "z = -2.400.", "meta": {"id": "WUA_STAT_25K_014593", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.98, s=17.22, n=34, compute CI."}]}, "response": "95% CI ≈ [53.19, 64.77].", "meta": {"id": "WUA_STAT_25K_014594", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.6, s=31.64, n=132, compute CI."}]}, "response": "95% CI ≈ [81.20, 92.00].", "meta": {"id": "WUA_STAT_25K_014595", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.11, sens=0.71, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.403.", "meta": {"id": "WUA_STAT_25K_014596", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.18, μ=1.26, σ=22.31, compute z."}]}, "response": "z = 0.714.", "meta": {"id": "WUA_STAT_25K_014597", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=169.27, s=8.59, n=11, compute CI."}]}, "response": "95% CI ≈ [164.19, 174.35].", "meta": {"id": "WUA_STAT_25K_014598", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.57, compute E and Var."}]}, "response": "E[X]=20.52; Var[X]=8.824.", "meta": {"id": "WUA_STAT_25K_014599", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.15, compute E and Var."}]}, "response": "E[X]=5.85; Var[X]=4.972.", "meta": {"id": "WUA_STAT_25K_014600", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.39, μ=55.21, σ=6.53, compute z."}]}, "response": "z = -3.035.", "meta": {"id": "WUA_STAT_25K_014601", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.31, s=16.41, n=11, compute CI."}]}, "response": "95% CI ≈ [40.61, 60.01].", "meta": {"id": "WUA_STAT_25K_014602", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.75, s=6.15, n=145, compute CI."}]}, "response": "95% CI ≈ [77.75, 79.75].", "meta": {"id": "WUA_STAT_25K_014603", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.044, sens=0.84, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.244.", "meta": {"id": "WUA_STAT_25K_014604", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.69, compute E and Var."}]}, "response": "E[X]=15.87; Var[X]=4.920.", "meta": {"id": "WUA_STAT_25K_014605", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.8, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.583.", "meta": {"id": "WUA_STAT_25K_014606", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.046, sens=0.86, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.153.", "meta": {"id": "WUA_STAT_25K_014607", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.81, μ=95.48, σ=10.79, compute z."}]}, "response": "z = -7.476.", "meta": {"id": "WUA_STAT_25K_014608", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.153, sens=0.83, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.555.", "meta": {"id": "WUA_STAT_25K_014609", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=100.07, s=32.69, n=190, compute CI."}]}, "response": "95% CI ≈ [95.42, 104.72].", "meta": {"id": "WUA_STAT_25K_014610", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.89, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.689.", "meta": {"id": "WUA_STAT_25K_014611", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.37, compute E and Var."}]}, "response": "E[X]=2.59; Var[X]=1.632.", "meta": {"id": "WUA_STAT_25K_014612", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.83, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.281.", "meta": {"id": "WUA_STAT_25K_014613", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.6, s=24.33, n=190, compute CI."}]}, "response": "95% CI ≈ [184.14, 191.06].", "meta": {"id": "WUA_STAT_25K_014614", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.27, μ=38.14, σ=1.73, compute z."}]}, "response": "z = 16.260.", "meta": {"id": "WUA_STAT_25K_014615", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.01, s=4.53, n=191, compute CI."}]}, "response": "95% CI ≈ [28.37, 29.65].", "meta": {"id": "WUA_STAT_25K_014616", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.28, compute E and Var."}]}, "response": "E[X]=4.48; Var[X]=3.226.", "meta": {"id": "WUA_STAT_25K_014617", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.119, sens=0.96, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.499.", "meta": {"id": "WUA_STAT_25K_014618", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.91, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.469.", "meta": {"id": "WUA_STAT_25K_014619", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.55, s=3.82, n=142, compute CI."}]}, "response": "95% CI ≈ [86.92, 88.18].", "meta": {"id": "WUA_STAT_25K_014620", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.14, μ=16.08, σ=16.61, compute z."}]}, "response": "z = 0.425.", "meta": {"id": "WUA_STAT_25K_014621", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.18, μ=34.08, σ=19.69, compute z."}]}, "response": "z = 1.478.", "meta": {"id": "WUA_STAT_25K_014622", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.4, μ=64.54, σ=5.82, compute z."}]}, "response": "z = -4.148.", "meta": {"id": "WUA_STAT_25K_014623", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.65, compute E and Var."}]}, "response": "E[X]=24.70; Var[X]=8.645.", "meta": {"id": "WUA_STAT_25K_014624", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.79, μ=50.3, σ=22.52, compute z."}]}, "response": "z = 1.398.", "meta": {"id": "WUA_STAT_25K_014625", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.7, μ=78.11, σ=12.91, compute z."}]}, "response": "z = -3.750.", "meta": {"id": "WUA_STAT_25K_014626", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=165.34, s=23.05, n=36, compute CI."}]}, "response": "95% CI ≈ [157.81, 172.87].", "meta": {"id": "WUA_STAT_25K_014627", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.051, sens=0.91, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.164.", "meta": {"id": "WUA_STAT_25K_014628", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.76, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.068.", "meta": {"id": "WUA_STAT_25K_014629", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.73, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.518.", "meta": {"id": "WUA_STAT_25K_014630", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.37, compute E and Var."}]}, "response": "E[X]=7.03; Var[X]=4.429.", "meta": {"id": "WUA_STAT_25K_014631", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.159, sens=0.89, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.359.", "meta": {"id": "WUA_STAT_25K_014632", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.98, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.714.", "meta": {"id": "WUA_STAT_25K_014633", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.26, μ=54.94, σ=20.11, compute z."}]}, "response": "z = 1.906.", "meta": {"id": "WUA_STAT_25K_014634", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.016, sens=0.85, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.090.", "meta": {"id": "WUA_STAT_25K_014635", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.042, sens=0.96, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.412.", "meta": {"id": "WUA_STAT_25K_014636", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.25, s=20.36, n=93, compute CI."}]}, "response": "95% CI ≈ [17.11, 25.39].", "meta": {"id": "WUA_STAT_25K_014637", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=149.88, s=33.76, n=170, compute CI."}]}, "response": "95% CI ≈ [144.81, 154.95].", "meta": {"id": "WUA_STAT_25K_014638", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.6, s=6.78, n=95, compute CI."}]}, "response": "95% CI ≈ [35.24, 37.96].", "meta": {"id": "WUA_STAT_25K_014639", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.56, μ=6.13, σ=8.98, compute z."}]}, "response": "z = 4.391.", "meta": {"id": "WUA_STAT_25K_014640", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=28.9, s=38.35, n=84, compute CI."}]}, "response": "95% CI ≈ [20.70, 37.10].", "meta": {"id": "WUA_STAT_25K_014641", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.224, sens=0.7, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.871.", "meta": {"id": "WUA_STAT_25K_014642", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.08, μ=51.6, σ=23.37, compute z."}]}, "response": "z = -1.905.", "meta": {"id": "WUA_STAT_25K_014643", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.9, s=29.59, n=146, compute CI."}]}, "response": "95% CI ≈ [186.10, 195.70].", "meta": {"id": "WUA_STAT_25K_014644", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.32, μ=24.27, σ=19.97, compute z."}]}, "response": "z = 1.955.", "meta": {"id": "WUA_STAT_25K_014645", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.213, sens=0.76, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.804.", "meta": {"id": "WUA_STAT_25K_014646", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.32, s=23.55, n=138, compute CI."}]}, "response": "95% CI ≈ [72.39, 80.25].", "meta": {"id": "WUA_STAT_25K_014647", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.49, s=34.41, n=163, compute CI."}]}, "response": "95% CI ≈ [54.21, 64.77].", "meta": {"id": "WUA_STAT_25K_014648", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.55, μ=70.49, σ=3.35, compute z."}]}, "response": "z = -9.534.", "meta": {"id": "WUA_STAT_25K_014649", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.054, sens=0.92, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.158.", "meta": {"id": "WUA_STAT_25K_014650", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.026, sens=0.8, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.176.", "meta": {"id": "WUA_STAT_25K_014651", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.221, sens=0.89, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.808.", "meta": {"id": "WUA_STAT_25K_014652", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.1, μ=76.84, σ=1.28, compute z."}]}, "response": "z = 0.203.", "meta": {"id": "WUA_STAT_25K_014653", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.201, sens=0.98, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.892.", "meta": {"id": "WUA_STAT_25K_014654", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.95, μ=36.4, σ=15.5, compute z."}]}, "response": "z = 1.390.", "meta": {"id": "WUA_STAT_25K_014655", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.97, μ=59.15, σ=5.56, compute z."}]}, "response": "z = 5.903.", "meta": {"id": "WUA_STAT_25K_014656", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.24, compute E and Var."}]}, "response": "E[X]=2.16; Var[X]=1.642.", "meta": {"id": "WUA_STAT_25K_014657", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.73, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.371.", "meta": {"id": "WUA_STAT_25K_014658", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.7, μ=18.59, σ=11.08, compute z."}]}, "response": "z = 5.696.", "meta": {"id": "WUA_STAT_25K_014659", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.1, compute E and Var."}]}, "response": "E[X]=2.40; Var[X]=2.160.", "meta": {"id": "WUA_STAT_25K_014660", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.47, μ=57.61, σ=8.66, compute z."}]}, "response": "z = 3.102.", "meta": {"id": "WUA_STAT_25K_014661", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.05, μ=23.93, σ=14.41, compute z."}]}, "response": "z = 4.033.", "meta": {"id": "WUA_STAT_25K_014662", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.56, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=3.696.", "meta": {"id": "WUA_STAT_25K_014663", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.49, compute E and Var."}]}, "response": "E[X]=5.39; Var[X]=2.749.", "meta": {"id": "WUA_STAT_25K_014664", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.64, compute E and Var."}]}, "response": "E[X]=20.48; Var[X]=7.373.", "meta": {"id": "WUA_STAT_25K_014665", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.76, s=18.56, n=159, compute CI."}]}, "response": "95% CI ≈ [104.88, 110.64].", "meta": {"id": "WUA_STAT_25K_014666", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.51, μ=46.52, σ=9.33, compute z."}]}, "response": "z = 5.465.", "meta": {"id": "WUA_STAT_25K_014667", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.32, μ=32.05, σ=10.25, compute z."}]}, "response": "z = 5.002.", "meta": {"id": "WUA_STAT_25K_014668", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.55, compute E and Var."}]}, "response": "E[X]=15.40; Var[X]=6.930.", "meta": {"id": "WUA_STAT_25K_014669", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.27, s=36.75, n=170, compute CI."}]}, "response": "95% CI ≈ [56.75, 67.79].", "meta": {"id": "WUA_STAT_25K_014670", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.93, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.133.", "meta": {"id": "WUA_STAT_25K_014671", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.75, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.294.", "meta": {"id": "WUA_STAT_25K_014672", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.73, μ=21.33, σ=24.18, compute z."}]}, "response": "z = 2.208.", "meta": {"id": "WUA_STAT_25K_014673", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.085, sens=0.76, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.207.", "meta": {"id": "WUA_STAT_25K_014674", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.98, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.641.", "meta": {"id": "WUA_STAT_25K_014675", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.03, μ=69.52, σ=3.25, compute z."}]}, "response": "z = -15.228.", "meta": {"id": "WUA_STAT_25K_014676", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.24, compute E and Var."}]}, "response": "E[X]=3.60; Var[X]=2.736.", "meta": {"id": "WUA_STAT_25K_014677", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.83, s=22.64, n=124, compute CI."}]}, "response": "95% CI ≈ [72.85, 80.81].", "meta": {"id": "WUA_STAT_25K_014678", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.65, compute E and Var."}]}, "response": "E[X]=8.45; Var[X]=2.958.", "meta": {"id": "WUA_STAT_25K_014679", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.16, μ=38.29, σ=12.95, compute z."}]}, "response": "z = -1.323.", "meta": {"id": "WUA_STAT_25K_014680", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.4, compute E and Var."}]}, "response": "E[X]=6.80; Var[X]=4.080.", "meta": {"id": "WUA_STAT_25K_014681", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.81, μ=61.36, σ=14.05, compute z."}]}, "response": "z = -2.601.", "meta": {"id": "WUA_STAT_25K_014682", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.98, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.713.", "meta": {"id": "WUA_STAT_25K_014683", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.16, compute E and Var."}]}, "response": "E[X]=5.12; Var[X]=4.301.", "meta": {"id": "WUA_STAT_25K_014684", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=128.3, s=24.16, n=171, compute CI."}]}, "response": "95% CI ≈ [124.68, 131.92].", "meta": {"id": "WUA_STAT_25K_014685", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.92, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.721.", "meta": {"id": "WUA_STAT_25K_014686", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.24, compute E and Var."}]}, "response": "E[X]=6.96; Var[X]=5.290.", "meta": {"id": "WUA_STAT_25K_014687", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.71, compute E and Var."}]}, "response": "E[X]=15.62; Var[X]=4.530.", "meta": {"id": "WUA_STAT_25K_014688", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.26, s=23.08, n=165, compute CI."}]}, "response": "95% CI ≈ [119.74, 126.78].", "meta": {"id": "WUA_STAT_25K_014689", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.07, compute E and Var."}]}, "response": "E[X]=1.68; Var[X]=1.562.", "meta": {"id": "WUA_STAT_25K_014690", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.94, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.734.", "meta": {"id": "WUA_STAT_25K_014691", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=102.66, s=3.39, n=49, compute CI."}]}, "response": "95% CI ≈ [101.71, 103.61].", "meta": {"id": "WUA_STAT_25K_014692", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.07, compute E and Var."}]}, "response": "E[X]=1.75; Var[X]=1.628.", "meta": {"id": "WUA_STAT_25K_014693", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.87, s=7.89, n=162, compute CI."}]}, "response": "95% CI ≈ [172.66, 175.08].", "meta": {"id": "WUA_STAT_25K_014694", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.17, μ=66.27, σ=18.34, compute z."}]}, "response": "z = -0.442.", "meta": {"id": "WUA_STAT_25K_014695", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.66, s=23.18, n=23, compute CI."}]}, "response": "95% CI ≈ [103.19, 122.13].", "meta": {"id": "WUA_STAT_25K_014696", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.2, μ=90.79, σ=21.65, compute z."}]}, "response": "z = -0.397.", "meta": {"id": "WUA_STAT_25K_014697", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.53, μ=52.4, σ=9.29, compute z."}]}, "response": "z = -5.476.", "meta": {"id": "WUA_STAT_25K_014698", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.39, compute E and Var."}]}, "response": "E[X]=2.73; Var[X]=1.665.", "meta": {"id": "WUA_STAT_25K_014699", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.52, s=8.02, n=48, compute CI."}]}, "response": "95% CI ≈ [117.25, 121.79].", "meta": {"id": "WUA_STAT_25K_014700", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.26, compute E and Var."}]}, "response": "E[X]=9.88; Var[X]=7.311.", "meta": {"id": "WUA_STAT_25K_014701", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=149.39, s=23.24, n=95, compute CI."}]}, "response": "95% CI ≈ [144.72, 154.06].", "meta": {"id": "WUA_STAT_25K_014702", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.48, μ=60.0, σ=8.19, compute z."}]}, "response": "z = -2.628.", "meta": {"id": "WUA_STAT_25K_014703", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.69, compute E and Var."}]}, "response": "E[X]=4.14; Var[X]=1.283.", "meta": {"id": "WUA_STAT_25K_014704", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.9, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.966.", "meta": {"id": "WUA_STAT_25K_014705", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.213, sens=0.8, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.436.", "meta": {"id": "WUA_STAT_25K_014706", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.55, s=16.46, n=177, compute CI."}]}, "response": "95% CI ≈ [35.13, 39.97].", "meta": {"id": "WUA_STAT_25K_014707", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.79, s=26.75, n=198, compute CI."}]}, "response": "95% CI ≈ [76.06, 83.52].", "meta": {"id": "WUA_STAT_25K_014708", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.35, compute E and Var."}]}, "response": "E[X]=4.20; Var[X]=2.730.", "meta": {"id": "WUA_STAT_25K_014709", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.53, compute E and Var."}]}, "response": "E[X]=4.24; Var[X]=1.993.", "meta": {"id": "WUA_STAT_25K_014710", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.05, μ=46.14, σ=3.99, compute z."}]}, "response": "z = -2.278.", "meta": {"id": "WUA_STAT_25K_014711", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.9, s=19.95, n=75, compute CI."}]}, "response": "95% CI ≈ [94.38, 103.42].", "meta": {"id": "WUA_STAT_25K_014712", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.45, compute E and Var."}]}, "response": "E[X]=9.45; Var[X]=5.198.", "meta": {"id": "WUA_STAT_25K_014713", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.67, compute E and Var."}]}, "response": "E[X]=10.05; Var[X]=3.317.", "meta": {"id": "WUA_STAT_25K_014714", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.141, sens=0.74, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.669.", "meta": {"id": "WUA_STAT_25K_014715", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.54, μ=82.62, σ=10.83, compute z."}]}, "response": "z = -4.994.", "meta": {"id": "WUA_STAT_25K_014716", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.07, compute E and Var."}]}, "response": "E[X]=1.61; Var[X]=1.497.", "meta": {"id": "WUA_STAT_25K_014717", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.041, sens=0.83, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.283.", "meta": {"id": "WUA_STAT_25K_014718", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.64, compute E and Var."}]}, "response": "E[X]=20.48; Var[X]=7.373.", "meta": {"id": "WUA_STAT_25K_014719", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.249, sens=0.97, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.728.", "meta": {"id": "WUA_STAT_25K_014720", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.69, compute E and Var."}]}, "response": "E[X]=9.66; Var[X]=2.995.", "meta": {"id": "WUA_STAT_25K_014721", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.88, μ=48.65, σ=18.85, compute z."}]}, "response": "z = 2.718.", "meta": {"id": "WUA_STAT_25K_014722", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.62, μ=5.66, σ=17.21, compute z."}]}, "response": "z = 1.799.", "meta": {"id": "WUA_STAT_25K_014723", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.6, s=29.21, n=62, compute CI."}]}, "response": "95% CI ≈ [71.33, 85.87].", "meta": {"id": "WUA_STAT_25K_014724", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.27, μ=97.24, σ=14.88, compute z."}]}, "response": "z = -2.417.", "meta": {"id": "WUA_STAT_25K_014725", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.95, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.700.", "meta": {"id": "WUA_STAT_25K_014726", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.07, compute E and Var."}]}, "response": "E[X]=2.31; Var[X]=2.148.", "meta": {"id": "WUA_STAT_25K_014727", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.238, sens=0.83, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.464.", "meta": {"id": "WUA_STAT_25K_014728", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.7, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=1.680.", "meta": {"id": "WUA_STAT_25K_014729", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.07, μ=60.66, σ=5.81, compute z."}]}, "response": "z = -8.363.", "meta": {"id": "WUA_STAT_25K_014730", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.044, sens=0.87, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.143.", "meta": {"id": "WUA_STAT_25K_014731", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.0, μ=55.96, σ=21.54, compute z."}]}, "response": "z = 1.116.", "meta": {"id": "WUA_STAT_25K_014732", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.19, s=33.8, n=180, compute CI."}]}, "response": "95% CI ≈ [132.25, 142.13].", "meta": {"id": "WUA_STAT_25K_014733", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.141, sens=0.75, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.528.", "meta": {"id": "WUA_STAT_25K_014734", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.39, μ=66.25, σ=18.04, compute z."}]}, "response": "z = -2.154.", "meta": {"id": "WUA_STAT_25K_014735", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.051, sens=0.94, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.219.", "meta": {"id": "WUA_STAT_25K_014736", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.223, sens=0.98, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.758.", "meta": {"id": "WUA_STAT_25K_014737", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=82.02, s=19.16, n=58, compute CI."}]}, "response": "95% CI ≈ [77.09, 86.95].", "meta": {"id": "WUA_STAT_25K_014738", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.24, μ=30.9, σ=22.4, compute z."}]}, "response": "z = 0.774.", "meta": {"id": "WUA_STAT_25K_014739", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.8, s=2.84, n=151, compute CI."}]}, "response": "95% CI ≈ [101.35, 102.25].", "meta": {"id": "WUA_STAT_25K_014740", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.58, compute E and Var."}]}, "response": "E[X]=23.20; Var[X]=9.744.", "meta": {"id": "WUA_STAT_25K_014741", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.98, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.732.", "meta": {"id": "WUA_STAT_25K_014742", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.91, μ=79.99, σ=10.39, compute z."}]}, "response": "z = -6.937.", "meta": {"id": "WUA_STAT_25K_014743", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.71, compute E and Var."}]}, "response": "E[X]=14.20; Var[X]=4.118.", "meta": {"id": "WUA_STAT_25K_014744", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.068, sens=0.78, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.180.", "meta": {"id": "WUA_STAT_25K_014745", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.44, μ=55.26, σ=19.23, compute z."}]}, "response": "z = -1.447.", "meta": {"id": "WUA_STAT_25K_014746", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=197.16, s=22.65, n=51, compute CI."}]}, "response": "95% CI ≈ [190.94, 203.38].", "meta": {"id": "WUA_STAT_25K_014747", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.57, μ=56.33, σ=3.15, compute z."}]}, "response": "z = 3.251.", "meta": {"id": "WUA_STAT_25K_014748", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.75, compute E and Var."}]}, "response": "E[X]=7.50; Var[X]=1.875.", "meta": {"id": "WUA_STAT_25K_014749", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.77, compute E and Var."}]}, "response": "E[X]=13.86; Var[X]=3.188.", "meta": {"id": "WUA_STAT_25K_014750", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.44, μ=78.73, σ=17.14, compute z."}]}, "response": "z = -1.417.", "meta": {"id": "WUA_STAT_25K_014751", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.47, compute E and Var."}]}, "response": "E[X]=5.64; Var[X]=2.989.", "meta": {"id": "WUA_STAT_25K_014752", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.1, s=12.21, n=167, compute CI."}]}, "response": "95% CI ≈ [41.25, 44.95].", "meta": {"id": "WUA_STAT_25K_014753", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.45, μ=19.49, σ=13.9, compute z."}]}, "response": "z = -0.003.", "meta": {"id": "WUA_STAT_25K_014754", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.09, compute E and Var."}]}, "response": "E[X]=1.62; Var[X]=1.474.", "meta": {"id": "WUA_STAT_25K_014755", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.52, μ=97.85, σ=7.04, compute z."}]}, "response": "z = -10.700.", "meta": {"id": "WUA_STAT_25K_014756", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.097, sens=0.89, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.389.", "meta": {"id": "WUA_STAT_25K_014757", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.48, compute E and Var."}]}, "response": "E[X]=19.20; Var[X]=9.984.", "meta": {"id": "WUA_STAT_25K_014758", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.37, compute E and Var."}]}, "response": "E[X]=11.84; Var[X]=7.459.", "meta": {"id": "WUA_STAT_25K_014759", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=168.59, s=34.36, n=174, compute CI."}]}, "response": "95% CI ≈ [163.48, 173.70].", "meta": {"id": "WUA_STAT_25K_014760", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.64, compute E and Var."}]}, "response": "E[X]=12.16; Var[X]=4.378.", "meta": {"id": "WUA_STAT_25K_014761", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.19, s=22.29, n=100, compute CI."}]}, "response": "95% CI ≈ [185.82, 194.56].", "meta": {"id": "WUA_STAT_25K_014762", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.35, μ=16.81, σ=21.13, compute z."}]}, "response": "z = -0.779.", "meta": {"id": "WUA_STAT_25K_014763", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.83, μ=19.02, σ=18.63, compute z."}]}, "response": "z = 0.580.", "meta": {"id": "WUA_STAT_25K_014764", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.81, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.501.", "meta": {"id": "WUA_STAT_25K_014765", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.37, compute E and Var."}]}, "response": "E[X]=11.47; Var[X]=7.226.", "meta": {"id": "WUA_STAT_25K_014766", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.69, compute E and Var."}]}, "response": "E[X]=6.90; Var[X]=2.139.", "meta": {"id": "WUA_STAT_25K_014767", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.91, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.224.", "meta": {"id": "WUA_STAT_25K_014768", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.2, compute E and Var."}]}, "response": "E[X]=2.40; Var[X]=1.920.", "meta": {"id": "WUA_STAT_25K_014769", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.66, μ=62.75, σ=2.51, compute z."}]}, "response": "z = -6.410.", "meta": {"id": "WUA_STAT_25K_014770", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.71, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.176.", "meta": {"id": "WUA_STAT_25K_014771", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.37, s=25.24, n=98, compute CI."}]}, "response": "95% CI ≈ [165.37, 175.37].", "meta": {"id": "WUA_STAT_25K_014772", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.78, s=2.97, n=126, compute CI."}]}, "response": "95% CI ≈ [74.26, 75.30].", "meta": {"id": "WUA_STAT_25K_014773", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.07, compute E and Var."}]}, "response": "E[X]=2.80; Var[X]=2.604.", "meta": {"id": "WUA_STAT_25K_014774", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.42, μ=40.03, σ=17.36, compute z."}]}, "response": "z = 2.499.", "meta": {"id": "WUA_STAT_25K_014775", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=173.1, s=1.61, n=158, compute CI."}]}, "response": "95% CI ≈ [172.85, 173.35].", "meta": {"id": "WUA_STAT_25K_014776", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.0, s=8.35, n=52, compute CI."}]}, "response": "95% CI ≈ [60.73, 65.27].", "meta": {"id": "WUA_STAT_25K_014777", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.51, s=36.36, n=100, compute CI."}]}, "response": "95% CI ≈ [7.38, 21.64].", "meta": {"id": "WUA_STAT_25K_014778", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.97, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.798.", "meta": {"id": "WUA_STAT_25K_014779", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.99, s=14.94, n=21, compute CI."}]}, "response": "95% CI ≈ [47.60, 60.38].", "meta": {"id": "WUA_STAT_25K_014780", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.191, sens=0.78, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.551.", "meta": {"id": "WUA_STAT_25K_014781", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.03, μ=33.72, σ=15.71, compute z."}]}, "response": "z = 0.975.", "meta": {"id": "WUA_STAT_25K_014782", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.67, s=6.59, n=162, compute CI."}]}, "response": "95% CI ≈ [132.66, 134.68].", "meta": {"id": "WUA_STAT_25K_014783", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.34, s=2.76, n=97, compute CI."}]}, "response": "95% CI ≈ [34.79, 35.89].", "meta": {"id": "WUA_STAT_25K_014784", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.82, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.483.", "meta": {"id": "WUA_STAT_25K_014785", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.37, μ=91.85, σ=7.39, compute z."}]}, "response": "z = -4.666.", "meta": {"id": "WUA_STAT_25K_014786", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.173, sens=0.8, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.366.", "meta": {"id": "WUA_STAT_25K_014787", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.6, compute E and Var."}]}, "response": "E[X]=12.00; Var[X]=4.800.", "meta": {"id": "WUA_STAT_25K_014788", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.21, μ=6.67, σ=15.59, compute z."}]}, "response": "z = 2.023.", "meta": {"id": "WUA_STAT_25K_014789", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.98, s=7.7, n=158, compute CI."}]}, "response": "95% CI ≈ [132.78, 135.18].", "meta": {"id": "WUA_STAT_25K_014790", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.88, μ=4.48, σ=10.81, compute z."}]}, "response": "z = 2.165.", "meta": {"id": "WUA_STAT_25K_014791", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.93, μ=80.69, σ=14.69, compute z."}]}, "response": "z = 1.310.", "meta": {"id": "WUA_STAT_25K_014792", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.98, s=7.97, n=84, compute CI."}]}, "response": "95% CI ≈ [97.28, 100.68].", "meta": {"id": "WUA_STAT_25K_014793", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.128, sens=0.85, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.555.", "meta": {"id": "WUA_STAT_25K_014794", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=37.98, s=23.08, n=137, compute CI."}]}, "response": "95% CI ≈ [34.12, 41.84].", "meta": {"id": "WUA_STAT_25K_014795", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.79, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.305.", "meta": {"id": "WUA_STAT_25K_014796", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.94, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.236.", "meta": {"id": "WUA_STAT_25K_014797", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.53, compute E and Var."}]}, "response": "E[X]=10.07; Var[X]=4.733.", "meta": {"id": "WUA_STAT_25K_014798", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.98, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.758.", "meta": {"id": "WUA_STAT_25K_014799", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.58, compute E and Var."}]}, "response": "E[X]=19.72; Var[X]=8.282.", "meta": {"id": "WUA_STAT_25K_014800", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.97, s=4.92, n=138, compute CI."}]}, "response": "95% CI ≈ [14.15, 15.79].", "meta": {"id": "WUA_STAT_25K_014801", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.34, s=39.49, n=119, compute CI."}]}, "response": "95% CI ≈ [160.24, 174.44].", "meta": {"id": "WUA_STAT_25K_014802", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.3, compute E and Var."}]}, "response": "E[X]=10.50; Var[X]=7.350.", "meta": {"id": "WUA_STAT_25K_014803", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.168, sens=0.77, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.357.", "meta": {"id": "WUA_STAT_25K_014804", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.175, sens=0.77, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.405.", "meta": {"id": "WUA_STAT_25K_014805", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.05, μ=20.78, σ=23.08, compute z."}]}, "response": "z = 0.878.", "meta": {"id": "WUA_STAT_25K_014806", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.51, compute E and Var."}]}, "response": "E[X]=16.83; Var[X]=8.247.", "meta": {"id": "WUA_STAT_25K_014807", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.84, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.392.", "meta": {"id": "WUA_STAT_25K_014808", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.037, sens=0.81, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.119.", "meta": {"id": "WUA_STAT_25K_014809", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.54, μ=17.48, σ=18.07, compute z."}]}, "response": "z = -0.605.", "meta": {"id": "WUA_STAT_25K_014810", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.39, compute E and Var."}]}, "response": "E[X]=13.65; Var[X]=8.326.", "meta": {"id": "WUA_STAT_25K_014811", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.5, μ=98.4, σ=2.87, compute z."}]}, "response": "z = -6.585.", "meta": {"id": "WUA_STAT_25K_014812", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.11, s=14.86, n=61, compute CI."}]}, "response": "95% CI ≈ [162.38, 169.84].", "meta": {"id": "WUA_STAT_25K_014813", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.66, s=32.81, n=182, compute CI."}]}, "response": "95% CI ≈ [185.89, 195.43].", "meta": {"id": "WUA_STAT_25K_014814", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=116.9, s=18.64, n=37, compute CI."}]}, "response": "95% CI ≈ [110.89, 122.91].", "meta": {"id": "WUA_STAT_25K_014815", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.8, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.441.", "meta": {"id": "WUA_STAT_25K_014816", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.88, μ=82.52, σ=11.23, compute z."}]}, "response": "z = -2.817.", "meta": {"id": "WUA_STAT_25K_014817", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.31, compute E and Var."}]}, "response": "E[X]=4.65; Var[X]=3.208.", "meta": {"id": "WUA_STAT_25K_014818", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.76, s=20.93, n=62, compute CI."}]}, "response": "95% CI ≈ [13.55, 23.97].", "meta": {"id": "WUA_STAT_25K_014819", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.27, s=3.4, n=192, compute CI."}]}, "response": "95% CI ≈ [193.79, 194.75].", "meta": {"id": "WUA_STAT_25K_014820", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.16, s=32.59, n=197, compute CI."}]}, "response": "95% CI ≈ [69.61, 78.71].", "meta": {"id": "WUA_STAT_25K_014821", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.07, μ=77.87, σ=4.41, compute z."}]}, "response": "z = 2.993.", "meta": {"id": "WUA_STAT_25K_014822", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.4, compute E and Var."}]}, "response": "E[X]=2.80; Var[X]=1.680.", "meta": {"id": "WUA_STAT_25K_014823", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.96, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.337.", "meta": {"id": "WUA_STAT_25K_014824", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.11, s=8.01, n=127, compute CI."}]}, "response": "95% CI ≈ [131.72, 134.50].", "meta": {"id": "WUA_STAT_25K_014825", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.197, sens=0.86, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.657.", "meta": {"id": "WUA_STAT_25K_014826", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.95, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.285.", "meta": {"id": "WUA_STAT_25K_014827", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.44, s=11.22, n=141, compute CI."}]}, "response": "95% CI ≈ [32.59, 36.29].", "meta": {"id": "WUA_STAT_25K_014828", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.47, μ=1.11, σ=23.57, compute z."}]}, "response": "z = 2.264.", "meta": {"id": "WUA_STAT_25K_014829", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.73, compute E and Var."}]}, "response": "E[X]=9.49; Var[X]=2.562.", "meta": {"id": "WUA_STAT_25K_014830", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.51, s=23.71, n=32, compute CI."}]}, "response": "95% CI ≈ [98.29, 114.73].", "meta": {"id": "WUA_STAT_25K_014831", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.42, s=7.76, n=24, compute CI."}]}, "response": "95% CI ≈ [94.32, 100.52].", "meta": {"id": "WUA_STAT_25K_014832", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.38, compute E and Var."}]}, "response": "E[X]=8.36; Var[X]=5.183.", "meta": {"id": "WUA_STAT_25K_014833", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.76, μ=56.09, σ=4.81, compute z."}]}, "response": "z = -9.840.", "meta": {"id": "WUA_STAT_25K_014834", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.24, compute E and Var."}]}, "response": "E[X]=4.32; Var[X]=3.283.", "meta": {"id": "WUA_STAT_25K_014835", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.23, sens=0.98, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.560.", "meta": {"id": "WUA_STAT_25K_014836", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.103, sens=0.88, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.336.", "meta": {"id": "WUA_STAT_25K_014837", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.42, μ=4.74, σ=10.3, compute z."}]}, "response": "z = 8.610.", "meta": {"id": "WUA_STAT_25K_014838", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.201, sens=0.93, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.854.", "meta": {"id": "WUA_STAT_25K_014839", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=126.3, s=35.21, n=34, compute CI."}]}, "response": "95% CI ≈ [114.46, 138.14].", "meta": {"id": "WUA_STAT_25K_014840", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.98, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.666.", "meta": {"id": "WUA_STAT_25K_014841", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.071, sens=0.75, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.589.", "meta": {"id": "WUA_STAT_25K_014842", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.137, sens=0.8, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.514.", "meta": {"id": "WUA_STAT_25K_014843", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.7, compute E and Var."}]}, "response": "E[X]=20.30; Var[X]=6.090.", "meta": {"id": "WUA_STAT_25K_014844", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.64, μ=48.25, σ=6.43, compute z."}]}, "response": "z = -3.827.", "meta": {"id": "WUA_STAT_25K_014845", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.26, compute E and Var."}]}, "response": "E[X]=5.46; Var[X]=4.040.", "meta": {"id": "WUA_STAT_25K_014846", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.192, sens=0.73, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.813.", "meta": {"id": "WUA_STAT_25K_014847", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.21, s=22.68, n=52, compute CI."}]}, "response": "95% CI ≈ [40.05, 52.37].", "meta": {"id": "WUA_STAT_25K_014848", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.19, compute E and Var."}]}, "response": "E[X]=3.61; Var[X]=2.924.", "meta": {"id": "WUA_STAT_25K_014849", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.51, μ=86.08, σ=17.63, compute z."}]}, "response": "z = -3.322.", "meta": {"id": "WUA_STAT_25K_014850", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.74, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.310.", "meta": {"id": "WUA_STAT_25K_014851", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.22, μ=7.45, σ=6.25, compute z."}]}, "response": "z = 13.403.", "meta": {"id": "WUA_STAT_25K_014852", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.66, compute E and Var."}]}, "response": "E[X]=20.46; Var[X]=6.956.", "meta": {"id": "WUA_STAT_25K_014853", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.24, compute E and Var."}]}, "response": "E[X]=4.80; Var[X]=3.648.", "meta": {"id": "WUA_STAT_25K_014854", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.23, compute E and Var."}]}, "response": "E[X]=4.14; Var[X]=3.188.", "meta": {"id": "WUA_STAT_25K_014855", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.16, μ=1.98, σ=13.21, compute z."}]}, "response": "z = 0.468.", "meta": {"id": "WUA_STAT_25K_014856", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.64, compute E and Var."}]}, "response": "E[X]=3.20; Var[X]=1.152.", "meta": {"id": "WUA_STAT_25K_014857", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.75, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.645.", "meta": {"id": "WUA_STAT_25K_014858", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.15, μ=11.96, σ=14.39, compute z."}]}, "response": "z = 5.781.", "meta": {"id": "WUA_STAT_25K_014859", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.79, s=18.09, n=164, compute CI."}]}, "response": "95% CI ≈ [33.02, 38.56].", "meta": {"id": "WUA_STAT_25K_014860", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.43, compute E and Var."}]}, "response": "E[X]=9.03; Var[X]=5.147.", "meta": {"id": "WUA_STAT_25K_014861", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.32, s=23.65, n=44, compute CI."}]}, "response": "95% CI ≈ [76.33, 90.31].", "meta": {"id": "WUA_STAT_25K_014862", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=124.18, s=32.5, n=57, compute CI."}]}, "response": "95% CI ≈ [115.74, 132.62].", "meta": {"id": "WUA_STAT_25K_014863", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.58, μ=48.13, σ=22.54, compute z."}]}, "response": "z = -1.222.", "meta": {"id": "WUA_STAT_25K_014864", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.156, sens=0.88, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.803.", "meta": {"id": "WUA_STAT_25K_014865", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.75, s=6.47, n=104, compute CI."}]}, "response": "95% CI ≈ [173.51, 175.99].", "meta": {"id": "WUA_STAT_25K_014866", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.29, s=20.77, n=45, compute CI."}]}, "response": "95% CI ≈ [39.22, 51.36].", "meta": {"id": "WUA_STAT_25K_014867", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.9, μ=18.67, σ=13.94, compute z."}]}, "response": "z = 3.460.", "meta": {"id": "WUA_STAT_25K_014868", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.011, sens=0.9, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.032.", "meta": {"id": "WUA_STAT_25K_014869", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.13, compute E and Var."}]}, "response": "E[X]=0.65; Var[X]=0.566.", "meta": {"id": "WUA_STAT_25K_014870", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.35, μ=98.9, σ=8.31, compute z."}]}, "response": "z = -8.129.", "meta": {"id": "WUA_STAT_25K_014871", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=197.1, s=35.87, n=96, compute CI."}]}, "response": "95% CI ≈ [189.92, 204.28].", "meta": {"id": "WUA_STAT_25K_014872", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.159, sens=0.88, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.675.", "meta": {"id": "WUA_STAT_25K_014873", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.25, compute E and Var."}]}, "response": "E[X]=5.50; Var[X]=4.125.", "meta": {"id": "WUA_STAT_25K_014874", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.046, sens=0.75, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.341.", "meta": {"id": "WUA_STAT_25K_014875", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.055, sens=0.89, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.270.", "meta": {"id": "WUA_STAT_25K_014876", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.17, compute E and Var."}]}, "response": "E[X]=1.70; Var[X]=1.411.", "meta": {"id": "WUA_STAT_25K_014877", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.52, compute E and Var."}]}, "response": "E[X]=3.12; Var[X]=1.498.", "meta": {"id": "WUA_STAT_25K_014878", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.9, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.071.", "meta": {"id": "WUA_STAT_25K_014879", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=82.76, s=36.69, n=168, compute CI."}]}, "response": "95% CI ≈ [77.21, 88.31].", "meta": {"id": "WUA_STAT_25K_014880", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.29, μ=48.59, σ=14.86, compute z."}]}, "response": "z = -0.962.", "meta": {"id": "WUA_STAT_25K_014881", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.07, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=2.344.", "meta": {"id": "WUA_STAT_25K_014882", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.89, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.208.", "meta": {"id": "WUA_STAT_25K_014883", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.05, μ=35.02, σ=3.06, compute z."}]}, "response": "z = 0.663.", "meta": {"id": "WUA_STAT_25K_014884", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.75, μ=45.0, σ=9.88, compute z."}]}, "response": "z = 3.416.", "meta": {"id": "WUA_STAT_25K_014885", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.83, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.816.", "meta": {"id": "WUA_STAT_25K_014886", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.108, sens=0.98, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.350.", "meta": {"id": "WUA_STAT_25K_014887", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.5, μ=14.58, σ=16.92, compute z."}]}, "response": "z = -0.832.", "meta": {"id": "WUA_STAT_25K_014888", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.72, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.653.", "meta": {"id": "WUA_STAT_25K_014889", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.28, μ=28.79, σ=19.13, compute z."}]}, "response": "z = 0.653.", "meta": {"id": "WUA_STAT_25K_014890", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.82, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.830.", "meta": {"id": "WUA_STAT_25K_014891", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.4, compute E and Var."}]}, "response": "E[X]=13.20; Var[X]=7.920.", "meta": {"id": "WUA_STAT_25K_014892", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.34, compute E and Var."}]}, "response": "E[X]=11.90; Var[X]=7.854.", "meta": {"id": "WUA_STAT_25K_014893", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.4, μ=47.13, σ=17.93, compute z."}]}, "response": "z = 2.859.", "meta": {"id": "WUA_STAT_25K_014894", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.68, compute E and Var."}]}, "response": "E[X]=24.48; Var[X]=7.834.", "meta": {"id": "WUA_STAT_25K_014895", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.76, compute E and Var."}]}, "response": "E[X]=21.28; Var[X]=5.107.", "meta": {"id": "WUA_STAT_25K_014896", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.87, s=25.15, n=131, compute CI."}]}, "response": "95% CI ≈ [18.56, 27.18].", "meta": {"id": "WUA_STAT_25K_014897", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.089, sens=0.76, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.304.", "meta": {"id": "WUA_STAT_25K_014898", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.79, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.617.", "meta": {"id": "WUA_STAT_25K_014899", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.51, compute E and Var."}]}, "response": "E[X]=18.87; Var[X]=9.246.", "meta": {"id": "WUA_STAT_25K_014900", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.52, compute E and Var."}]}, "response": "E[X]=5.72; Var[X]=2.746.", "meta": {"id": "WUA_STAT_25K_014901", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.16, compute E and Var."}]}, "response": "E[X]=4.96; Var[X]=4.166.", "meta": {"id": "WUA_STAT_25K_014902", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.34, s=38.95, n=104, compute CI."}]}, "response": "95% CI ≈ [66.85, 81.83].", "meta": {"id": "WUA_STAT_25K_014903", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.96, s=22.02, n=94, compute CI."}]}, "response": "95% CI ≈ [155.51, 164.41].", "meta": {"id": "WUA_STAT_25K_014904", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.054, sens=0.81, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.142.", "meta": {"id": "WUA_STAT_25K_014905", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.25, compute E and Var."}]}, "response": "E[X]=6.00; Var[X]=4.500.", "meta": {"id": "WUA_STAT_25K_014906", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.39, compute E and Var."}]}, "response": "E[X]=12.09; Var[X]=7.375.", "meta": {"id": "WUA_STAT_25K_014907", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.03, s=38.55, n=67, compute CI."}]}, "response": "95% CI ≈ [51.80, 70.26].", "meta": {"id": "WUA_STAT_25K_014908", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.13, compute E and Var."}]}, "response": "E[X]=5.07; Var[X]=4.411.", "meta": {"id": "WUA_STAT_25K_014909", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.02, s=19.9, n=156, compute CI."}]}, "response": "95% CI ≈ [153.90, 160.14].", "meta": {"id": "WUA_STAT_25K_014910", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.41, compute E and Var."}]}, "response": "E[X]=9.02; Var[X]=5.322.", "meta": {"id": "WUA_STAT_25K_014911", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.48, compute E and Var."}]}, "response": "E[X]=5.28; Var[X]=2.746.", "meta": {"id": "WUA_STAT_25K_014912", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.79, μ=82.29, σ=3.23, compute z."}]}, "response": "z = -12.229.", "meta": {"id": "WUA_STAT_25K_014913", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.67, μ=19.36, σ=2.89, compute z."}]}, "response": "z = 14.986.", "meta": {"id": "WUA_STAT_25K_014914", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.57, compute E and Var."}]}, "response": "E[X]=22.23; Var[X]=9.559.", "meta": {"id": "WUA_STAT_25K_014915", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.233, sens=0.84, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.927.", "meta": {"id": "WUA_STAT_25K_014916", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.24, compute E and Var."}]}, "response": "E[X]=7.92; Var[X]=6.019.", "meta": {"id": "WUA_STAT_25K_014917", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.54, compute E and Var."}]}, "response": "E[X]=9.72; Var[X]=4.471.", "meta": {"id": "WUA_STAT_25K_014918", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.78, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.302.", "meta": {"id": "WUA_STAT_25K_014919", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.65, s=34.29, n=126, compute CI."}]}, "response": "95% CI ≈ [188.66, 200.64].", "meta": {"id": "WUA_STAT_25K_014920", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.58, compute E and Var."}]}, "response": "E[X]=13.34; Var[X]=5.603.", "meta": {"id": "WUA_STAT_25K_014921", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.39, μ=92.58, σ=8.74, compute z."}]}, "response": "z = -1.852.", "meta": {"id": "WUA_STAT_25K_014922", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.23, compute E and Var."}]}, "response": "E[X]=3.91; Var[X]=3.011.", "meta": {"id": "WUA_STAT_25K_014923", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.76, μ=22.59, σ=18.11, compute z."}]}, "response": "z = 3.985.", "meta": {"id": "WUA_STAT_25K_014924", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.73, s=32.77, n=178, compute CI."}]}, "response": "95% CI ≈ [22.92, 32.54].", "meta": {"id": "WUA_STAT_25K_014925", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.9, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.474.", "meta": {"id": "WUA_STAT_25K_014926", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.6, compute E and Var."}]}, "response": "E[X]=15.60; Var[X]=6.240.", "meta": {"id": "WUA_STAT_25K_014927", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.09, compute E and Var."}]}, "response": "E[X]=1.26; Var[X]=1.147.", "meta": {"id": "WUA_STAT_25K_014928", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.101, sens=0.86, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.377.", "meta": {"id": "WUA_STAT_25K_014929", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.72, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.768.", "meta": {"id": "WUA_STAT_25K_014930", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.84, μ=61.39, σ=6.31, compute z."}]}, "response": "z = 5.777.", "meta": {"id": "WUA_STAT_25K_014931", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.19, compute E and Var."}]}, "response": "E[X]=6.08; Var[X]=4.925.", "meta": {"id": "WUA_STAT_25K_014932", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.51, s=26.03, n=58, compute CI."}]}, "response": "95% CI ≈ [6.81, 20.21].", "meta": {"id": "WUA_STAT_25K_014933", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.63, compute E and Var."}]}, "response": "E[X]=18.90; Var[X]=6.993.", "meta": {"id": "WUA_STAT_25K_014934", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.79, s=10.3, n=37, compute CI."}]}, "response": "95% CI ≈ [84.47, 91.11].", "meta": {"id": "WUA_STAT_25K_014935", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.02, sens=0.83, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.055.", "meta": {"id": "WUA_STAT_25K_014936", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.157, sens=0.79, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.620.", "meta": {"id": "WUA_STAT_25K_014937", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.36, s=2.29, n=178, compute CI."}]}, "response": "95% CI ≈ [30.02, 30.70].", "meta": {"id": "WUA_STAT_25K_014938", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.25, compute E and Var."}]}, "response": "E[X]=4.50; Var[X]=3.375.", "meta": {"id": "WUA_STAT_25K_014939", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.63, compute E and Var."}]}, "response": "E[X]=6.30; Var[X]=2.331.", "meta": {"id": "WUA_STAT_25K_014940", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=177.89, s=27.65, n=13, compute CI."}]}, "response": "95% CI ≈ [162.86, 192.92].", "meta": {"id": "WUA_STAT_25K_014941", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.57, s=4.83, n=13, compute CI."}]}, "response": "95% CI ≈ [81.94, 87.20].", "meta": {"id": "WUA_STAT_25K_014942", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.74, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.486.", "meta": {"id": "WUA_STAT_25K_014943", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.59, compute E and Var."}]}, "response": "E[X]=17.11; Var[X]=7.015.", "meta": {"id": "WUA_STAT_25K_014944", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.26, μ=47.53, σ=13.2, compute z."}]}, "response": "z = 0.964.", "meta": {"id": "WUA_STAT_25K_014945", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.28, compute E and Var."}]}, "response": "E[X]=1.96; Var[X]=1.411.", "meta": {"id": "WUA_STAT_25K_014946", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.56, μ=13.43, σ=19.15, compute z."}]}, "response": "z = 2.566.", "meta": {"id": "WUA_STAT_25K_014947", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.71, s=9.45, n=118, compute CI."}]}, "response": "95% CI ≈ [74.00, 77.42].", "meta": {"id": "WUA_STAT_25K_014948", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.78, compute E and Var."}]}, "response": "E[X]=10.14; Var[X]=2.231.", "meta": {"id": "WUA_STAT_25K_014949", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=195.36, s=22.49, n=51, compute CI."}]}, "response": "95% CI ≈ [189.19, 201.53].", "meta": {"id": "WUA_STAT_25K_014950", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.12, μ=10.58, σ=13.0, compute z."}]}, "response": "z = 1.580.", "meta": {"id": "WUA_STAT_25K_014951", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.62, compute E and Var."}]}, "response": "E[X]=5.58; Var[X]=2.120.", "meta": {"id": "WUA_STAT_25K_014952", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.61, s=4.57, n=168, compute CI."}]}, "response": "95% CI ≈ [62.92, 64.30].", "meta": {"id": "WUA_STAT_25K_014953", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.122, sens=0.82, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.388.", "meta": {"id": "WUA_STAT_25K_014954", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.18, μ=12.56, σ=15.77, compute z."}]}, "response": "z = -0.595.", "meta": {"id": "WUA_STAT_25K_014955", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.77, s=22.42, n=152, compute CI."}]}, "response": "95% CI ≈ [42.21, 49.33].", "meta": {"id": "WUA_STAT_25K_014956", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.8, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.394.", "meta": {"id": "WUA_STAT_25K_014957", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.8, compute E and Var."}]}, "response": "E[X]=29.60; Var[X]=5.920.", "meta": {"id": "WUA_STAT_25K_014958", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.46, s=24.21, n=131, compute CI."}]}, "response": "95% CI ≈ [8.31, 16.61].", "meta": {"id": "WUA_STAT_25K_014959", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.35, compute E and Var."}]}, "response": "E[X]=4.55; Var[X]=2.958.", "meta": {"id": "WUA_STAT_25K_014960", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.211, sens=0.84, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.529.", "meta": {"id": "WUA_STAT_25K_014961", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=191.25, s=12.75, n=100, compute CI."}]}, "response": "95% CI ≈ [188.75, 193.75].", "meta": {"id": "WUA_STAT_25K_014962", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.72, compute E and Var."}]}, "response": "E[X]=10.80; Var[X]=3.024.", "meta": {"id": "WUA_STAT_25K_014963", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.3, compute E and Var."}]}, "response": "E[X]=1.80; Var[X]=1.260.", "meta": {"id": "WUA_STAT_25K_014964", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.93, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.771.", "meta": {"id": "WUA_STAT_25K_014965", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.84, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.181.", "meta": {"id": "WUA_STAT_25K_014966", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.37, μ=77.75, σ=22.84, compute z."}]}, "response": "z = -2.731.", "meta": {"id": "WUA_STAT_25K_014967", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.06, s=24.29, n=85, compute CI."}]}, "response": "95% CI ≈ [146.90, 157.22].", "meta": {"id": "WUA_STAT_25K_014968", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.85, μ=41.92, σ=14.67, compute z."}]}, "response": "z = 1.086.", "meta": {"id": "WUA_STAT_25K_014969", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=26.72, s=39.21, n=57, compute CI."}]}, "response": "95% CI ≈ [16.54, 36.90].", "meta": {"id": "WUA_STAT_25K_014970", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.99, μ=72.88, σ=20.51, compute z."}]}, "response": "z = -0.726.", "meta": {"id": "WUA_STAT_25K_014971", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.43, s=7.45, n=25, compute CI."}]}, "response": "95% CI ≈ [10.51, 16.35].", "meta": {"id": "WUA_STAT_25K_014972", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.2, μ=10.67, σ=13.19, compute z."}]}, "response": "z = 2.542.", "meta": {"id": "WUA_STAT_25K_014973", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.216, sens=0.78, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.443.", "meta": {"id": "WUA_STAT_25K_014974", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.61, μ=30.55, σ=5.18, compute z."}]}, "response": "z = -4.622.", "meta": {"id": "WUA_STAT_25K_014975", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=196.95, s=6.84, n=196, compute CI."}]}, "response": "95% CI ≈ [195.99, 197.91].", "meta": {"id": "WUA_STAT_25K_014976", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.64, compute E and Var."}]}, "response": "E[X]=12.80; Var[X]=4.608.", "meta": {"id": "WUA_STAT_25K_014977", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.42, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.218.", "meta": {"id": "WUA_STAT_25K_014978", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.7, μ=71.47, σ=1.71, compute z."}]}, "response": "z = -31.444.", "meta": {"id": "WUA_STAT_25K_014979", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.68, s=10.76, n=169, compute CI."}]}, "response": "95% CI ≈ [49.06, 52.30].", "meta": {"id": "WUA_STAT_25K_014980", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.58, compute E and Var."}]}, "response": "E[X]=11.02; Var[X]=4.628.", "meta": {"id": "WUA_STAT_25K_014981", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.74, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.098.", "meta": {"id": "WUA_STAT_25K_014982", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=100.38, s=31.21, n=143, compute CI."}]}, "response": "95% CI ≈ [95.26, 105.50].", "meta": {"id": "WUA_STAT_25K_014983", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.41, s=3.69, n=130, compute CI."}]}, "response": "95% CI ≈ [53.78, 55.04].", "meta": {"id": "WUA_STAT_25K_014984", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.205, sens=0.87, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.494.", "meta": {"id": "WUA_STAT_25K_014985", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.241, sens=0.98, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.912.", "meta": {"id": "WUA_STAT_25K_014986", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.55, compute E and Var."}]}, "response": "E[X]=3.30; Var[X]=1.485.", "meta": {"id": "WUA_STAT_25K_014987", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.22, s=25.1, n=169, compute CI."}]}, "response": "95% CI ≈ [42.44, 50.00].", "meta": {"id": "WUA_STAT_25K_014988", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.3, compute E and Var."}]}, "response": "E[X]=3.30; Var[X]=2.310.", "meta": {"id": "WUA_STAT_25K_014989", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.61, μ=19.29, σ=15.32, compute z."}]}, "response": "z = 4.590.", "meta": {"id": "WUA_STAT_25K_014990", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.74, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.042.", "meta": {"id": "WUA_STAT_25K_014991", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.45, compute E and Var."}]}, "response": "E[X]=9.90; Var[X]=5.445.", "meta": {"id": "WUA_STAT_25K_014992", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=16.63, s=35.54, n=162, compute CI."}]}, "response": "95% CI ≈ [11.16, 22.10].", "meta": {"id": "WUA_STAT_25K_014993", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.41, compute E and Var."}]}, "response": "E[X]=2.46; Var[X]=1.451.", "meta": {"id": "WUA_STAT_25K_014994", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.77, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.580.", "meta": {"id": "WUA_STAT_25K_014995", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.3, compute E and Var."}]}, "response": "E[X]=10.80; Var[X]=7.560.", "meta": {"id": "WUA_STAT_25K_014996", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=162.7, s=25.63, n=105, compute CI."}]}, "response": "95% CI ≈ [157.80, 167.60].", "meta": {"id": "WUA_STAT_25K_014997", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.12, compute E and Var."}]}, "response": "E[X]=1.20; Var[X]=1.056.", "meta": {"id": "WUA_STAT_25K_014998", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.2, μ=71.46, σ=1.15, compute z."}]}, "response": "z = 7.600.", "meta": {"id": "WUA_STAT_25K_014999", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.7, μ=0.54, σ=24.26, compute z."}]}, "response": "z = 0.213.", "meta": {"id": "WUA_STAT_25K_015000", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.23, compute E and Var."}]}, "response": "E[X]=2.53; Var[X]=1.948.", "meta": {"id": "WUA_STAT_25K_015001", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.39, μ=59.9, σ=15.14, compute z."}]}, "response": "z = 2.476.", "meta": {"id": "WUA_STAT_25K_015002", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.76, compute E and Var."}]}, "response": "E[X]=17.48; Var[X]=4.195.", "meta": {"id": "WUA_STAT_25K_015003", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.78, compute E and Var."}]}, "response": "E[X]=4.68; Var[X]=1.030.", "meta": {"id": "WUA_STAT_25K_015004", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.031, sens=0.94, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.177.", "meta": {"id": "WUA_STAT_25K_015005", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.45, μ=27.25, σ=14.09, compute z."}]}, "response": "z = -0.554.", "meta": {"id": "WUA_STAT_25K_015006", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.229, sens=0.94, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.548.", "meta": {"id": "WUA_STAT_25K_015007", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.1, compute E and Var."}]}, "response": "E[X]=1.10; Var[X]=0.990.", "meta": {"id": "WUA_STAT_25K_015008", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.32, compute E and Var."}]}, "response": "E[X]=5.12; Var[X]=3.482.", "meta": {"id": "WUA_STAT_25K_015009", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.039, sens=0.72, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.092.", "meta": {"id": "WUA_STAT_25K_015010", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.21, μ=68.3, σ=10.91, compute z."}]}, "response": "z = 1.458.", "meta": {"id": "WUA_STAT_25K_015011", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.46, compute E and Var."}]}, "response": "E[X]=14.26; Var[X]=7.700.", "meta": {"id": "WUA_STAT_25K_015012", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=103.69, s=38.56, n=141, compute CI."}]}, "response": "95% CI ≈ [97.33, 110.05].", "meta": {"id": "WUA_STAT_25K_015013", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.58, compute E and Var."}]}, "response": "E[X]=8.70; Var[X]=3.654.", "meta": {"id": "WUA_STAT_25K_015014", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.74, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.679.", "meta": {"id": "WUA_STAT_25K_015015", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=114.4, s=38.31, n=161, compute CI."}]}, "response": "95% CI ≈ [108.48, 120.32].", "meta": {"id": "WUA_STAT_25K_015016", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.1, s=27.03, n=169, compute CI."}]}, "response": "95% CI ≈ [93.02, 101.18].", "meta": {"id": "WUA_STAT_25K_015017", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.14, sens=0.71, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.405.", "meta": {"id": "WUA_STAT_25K_015018", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.75, μ=29.03, σ=17.08, compute z."}]}, "response": "z = -0.953.", "meta": {"id": "WUA_STAT_25K_015019", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.72, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.291.", "meta": {"id": "WUA_STAT_25K_015020", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.78, s=11.35, n=104, compute CI."}]}, "response": "95% CI ≈ [46.60, 50.96].", "meta": {"id": "WUA_STAT_25K_015021", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.65, s=33.7, n=63, compute CI."}]}, "response": "95% CI ≈ [23.33, 39.97].", "meta": {"id": "WUA_STAT_25K_015022", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=102.07, s=33.55, n=28, compute CI."}]}, "response": "95% CI ≈ [89.64, 114.50].", "meta": {"id": "WUA_STAT_25K_015023", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.226, sens=0.91, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.610.", "meta": {"id": "WUA_STAT_25K_015024", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.3, compute E and Var."}]}, "response": "E[X]=4.50; Var[X]=3.150.", "meta": {"id": "WUA_STAT_25K_015025", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.25, compute E and Var."}]}, "response": "E[X]=8.50; Var[X]=6.375.", "meta": {"id": "WUA_STAT_25K_015026", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.22, s=27.98, n=191, compute CI."}]}, "response": "95% CI ≈ [7.25, 15.19].", "meta": {"id": "WUA_STAT_25K_015027", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.07, μ=84.17, σ=13.29, compute z."}]}, "response": "z = -5.726.", "meta": {"id": "WUA_STAT_25K_015028", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.71, μ=45.62, σ=8.67, compute z."}]}, "response": "z = 4.739.", "meta": {"id": "WUA_STAT_25K_015029", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.74, μ=32.72, σ=16.03, compute z."}]}, "response": "z = 1.187.", "meta": {"id": "WUA_STAT_25K_015030", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.54, s=31.39, n=111, compute CI."}]}, "response": "95% CI ≈ [105.70, 117.38].", "meta": {"id": "WUA_STAT_25K_015031", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.82, s=13.24, n=22, compute CI."}]}, "response": "95% CI ≈ [104.29, 115.35].", "meta": {"id": "WUA_STAT_25K_015032", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.57, μ=18.47, σ=2.56, compute z."}]}, "response": "z = -5.039.", "meta": {"id": "WUA_STAT_25K_015033", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.17, sens=0.81, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.624.", "meta": {"id": "WUA_STAT_25K_015034", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.35, s=15.74, n=125, compute CI."}]}, "response": "95% CI ≈ [154.59, 160.11].", "meta": {"id": "WUA_STAT_25K_015035", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.31, compute E and Var."}]}, "response": "E[X]=5.89; Var[X]=4.064.", "meta": {"id": "WUA_STAT_25K_015036", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.61, compute E and Var."}]}, "response": "E[X]=7.32; Var[X]=2.855.", "meta": {"id": "WUA_STAT_25K_015037", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.08, compute E and Var."}]}, "response": "E[X]=2.24; Var[X]=2.061.", "meta": {"id": "WUA_STAT_25K_015038", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.33, μ=81.42, σ=19.58, compute z."}]}, "response": "z = -0.515.", "meta": {"id": "WUA_STAT_25K_015039", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.61, s=28.64, n=72, compute CI."}]}, "response": "95% CI ≈ [51.99, 65.23].", "meta": {"id": "WUA_STAT_25K_015040", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.11, s=34.61, n=83, compute CI."}]}, "response": "95% CI ≈ [23.66, 38.56].", "meta": {"id": "WUA_STAT_25K_015041", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.77, μ=49.69, σ=17.73, compute z."}]}, "response": "z = 0.963.", "meta": {"id": "WUA_STAT_25K_015042", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.49, compute E and Var."}]}, "response": "E[X]=16.17; Var[X]=8.247.", "meta": {"id": "WUA_STAT_25K_015043", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.57, compute E and Var."}]}, "response": "E[X]=16.53; Var[X]=7.108.", "meta": {"id": "WUA_STAT_25K_015044", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.97, s=25.68, n=48, compute CI."}]}, "response": "95% CI ≈ [183.71, 198.23].", "meta": {"id": "WUA_STAT_25K_015045", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.47, compute E and Var."}]}, "response": "E[X]=8.93; Var[X]=4.733.", "meta": {"id": "WUA_STAT_25K_015046", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.85, μ=25.18, σ=8.6, compute z."}]}, "response": "z = 2.520.", "meta": {"id": "WUA_STAT_25K_015047", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.67, μ=59.19, σ=20.92, compute z."}]}, "response": "z = -1.889.", "meta": {"id": "WUA_STAT_25K_015048", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.75, μ=8.07, σ=16.0, compute z."}]}, "response": "z = 4.293.", "meta": {"id": "WUA_STAT_25K_015049", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.4, s=12.9, n=197, compute CI."}]}, "response": "95% CI ≈ [41.60, 45.20].", "meta": {"id": "WUA_STAT_25K_015050", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.3, compute E and Var."}]}, "response": "E[X]=10.50; Var[X]=7.350.", "meta": {"id": "WUA_STAT_25K_015051", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.56, μ=66.19, σ=14.95, compute z."}]}, "response": "z = -1.447.", "meta": {"id": "WUA_STAT_25K_015052", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=133.67, s=30.64, n=194, compute CI."}]}, "response": "95% CI ≈ [129.36, 137.98].", "meta": {"id": "WUA_STAT_25K_015053", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.66, compute E and Var."}]}, "response": "E[X]=16.50; Var[X]=5.610.", "meta": {"id": "WUA_STAT_25K_015054", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.1, s=32.45, n=72, compute CI."}]}, "response": "95% CI ≈ [185.60, 200.60].", "meta": {"id": "WUA_STAT_25K_015055", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.26, s=7.57, n=111, compute CI."}]}, "response": "95% CI ≈ [130.85, 133.67].", "meta": {"id": "WUA_STAT_25K_015056", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.35, s=4.08, n=58, compute CI."}]}, "response": "95% CI ≈ [13.30, 15.40].", "meta": {"id": "WUA_STAT_25K_015057", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.54, s=29.27, n=146, compute CI."}]}, "response": "95% CI ≈ [124.79, 134.29].", "meta": {"id": "WUA_STAT_25K_015058", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.21, compute E and Var."}]}, "response": "E[X]=1.26; Var[X]=0.995.", "meta": {"id": "WUA_STAT_25K_015059", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.34, μ=10.75, σ=10.68, compute z."}]}, "response": "z = 2.958.", "meta": {"id": "WUA_STAT_25K_015060", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.33, compute E and Var."}]}, "response": "E[X]=8.58; Var[X]=5.749.", "meta": {"id": "WUA_STAT_25K_015061", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.08, s=2.31, n=142, compute CI."}]}, "response": "95% CI ≈ [40.70, 41.46].", "meta": {"id": "WUA_STAT_25K_015062", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.38, μ=77.78, σ=2.79, compute z."}]}, "response": "z = -3.369.", "meta": {"id": "WUA_STAT_25K_015063", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.82, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.306.", "meta": {"id": "WUA_STAT_25K_015064", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.14, compute E and Var."}]}, "response": "E[X]=0.98; Var[X]=0.843.", "meta": {"id": "WUA_STAT_25K_015065", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.76, s=35.76, n=162, compute CI."}]}, "response": "95% CI ≈ [171.25, 182.27].", "meta": {"id": "WUA_STAT_25K_015066", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.39, compute E and Var."}]}, "response": "E[X]=10.14; Var[X]=6.185.", "meta": {"id": "WUA_STAT_25K_015067", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.42, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=1.462.", "meta": {"id": "WUA_STAT_25K_015068", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.97, s=4.52, n=34, compute CI."}]}, "response": "95% CI ≈ [76.45, 79.49].", "meta": {"id": "WUA_STAT_25K_015069", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.84, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.229.", "meta": {"id": "WUA_STAT_25K_015070", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=32.53, s=31.29, n=35, compute CI."}]}, "response": "95% CI ≈ [22.16, 42.90].", "meta": {"id": "WUA_STAT_25K_015071", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.77, compute E and Var."}]}, "response": "E[X]=19.25; Var[X]=4.427.", "meta": {"id": "WUA_STAT_25K_015072", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.55, compute E and Var."}]}, "response": "E[X]=2.75; Var[X]=1.237.", "meta": {"id": "WUA_STAT_25K_015073", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.17, compute E and Var."}]}, "response": "E[X]=1.19; Var[X]=0.988.", "meta": {"id": "WUA_STAT_25K_015074", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.23, s=15.76, n=38, compute CI."}]}, "response": "95% CI ≈ [179.22, 189.24].", "meta": {"id": "WUA_STAT_25K_015075", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.9, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.289.", "meta": {"id": "WUA_STAT_25K_015076", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.05, compute E and Var."}]}, "response": "E[X]=2.00; Var[X]=1.900.", "meta": {"id": "WUA_STAT_25K_015077", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.6, μ=48.74, σ=4.78, compute z."}]}, "response": "z = 3.109.", "meta": {"id": "WUA_STAT_25K_015078", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=47.82, s=38.13, n=22, compute CI."}]}, "response": "95% CI ≈ [31.89, 63.75].", "meta": {"id": "WUA_STAT_25K_015079", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.22, s=30.31, n=98, compute CI."}]}, "response": "95% CI ≈ [19.22, 31.22].", "meta": {"id": "WUA_STAT_25K_015080", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.05, μ=63.53, σ=5.16, compute z."}]}, "response": "z = 3.977.", "meta": {"id": "WUA_STAT_25K_015081", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.08, μ=67.61, σ=2.95, compute z."}]}, "response": "z = -8.315.", "meta": {"id": "WUA_STAT_25K_015082", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.96, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.606.", "meta": {"id": "WUA_STAT_25K_015083", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.121, sens=0.98, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.403.", "meta": {"id": "WUA_STAT_25K_015084", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=149.91, s=24.19, n=125, compute CI."}]}, "response": "95% CI ≈ [145.67, 154.15].", "meta": {"id": "WUA_STAT_25K_015085", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.38, compute E and Var."}]}, "response": "E[X]=3.80; Var[X]=2.356.", "meta": {"id": "WUA_STAT_25K_015086", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.45, μ=74.9, σ=1.49, compute z."}]}, "response": "z = -8.356.", "meta": {"id": "WUA_STAT_25K_015087", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.33, compute E and Var."}]}, "response": "E[X]=9.24; Var[X]=6.191.", "meta": {"id": "WUA_STAT_25K_015088", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.89, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.365.", "meta": {"id": "WUA_STAT_25K_015089", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.15, s=3.7, n=62, compute CI."}]}, "response": "95% CI ≈ [181.23, 183.07].", "meta": {"id": "WUA_STAT_25K_015090", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.67, compute E and Var."}]}, "response": "E[X]=12.06; Var[X]=3.980.", "meta": {"id": "WUA_STAT_25K_015091", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.24, compute E and Var."}]}, "response": "E[X]=5.28; Var[X]=4.013.", "meta": {"id": "WUA_STAT_25K_015092", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.54, μ=5.65, σ=15.74, compute z."}]}, "response": "z = 3.487.", "meta": {"id": "WUA_STAT_25K_015093", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.34, compute E and Var."}]}, "response": "E[X]=6.12; Var[X]=4.039.", "meta": {"id": "WUA_STAT_25K_015094", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.96, s=28.4, n=133, compute CI."}]}, "response": "95% CI ≈ [163.13, 172.79].", "meta": {"id": "WUA_STAT_25K_015095", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.064, sens=0.92, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.411.", "meta": {"id": "WUA_STAT_25K_015096", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.78, compute E and Var."}]}, "response": "E[X]=10.14; Var[X]=2.231.", "meta": {"id": "WUA_STAT_25K_015097", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.21, compute E and Var."}]}, "response": "E[X]=4.62; Var[X]=3.650.", "meta": {"id": "WUA_STAT_25K_015098", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.45, compute E and Var."}]}, "response": "E[X]=3.60; Var[X]=1.980.", "meta": {"id": "WUA_STAT_25K_015099", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.014, sens=0.92, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.106.", "meta": {"id": "WUA_STAT_25K_015100", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.4, compute E and Var."}]}, "response": "E[X]=10.80; Var[X]=6.480.", "meta": {"id": "WUA_STAT_25K_015101", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.63, compute E and Var."}]}, "response": "E[X]=13.86; Var[X]=5.128.", "meta": {"id": "WUA_STAT_25K_015102", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.87, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.377.", "meta": {"id": "WUA_STAT_25K_015103", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.73, compute E and Var."}]}, "response": "E[X]=25.55; Var[X]=6.899.", "meta": {"id": "WUA_STAT_25K_015104", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.96, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.433.", "meta": {"id": "WUA_STAT_25K_015105", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=10.0, s=21.04, n=145, compute CI."}]}, "response": "95% CI ≈ [6.58, 13.42].", "meta": {"id": "WUA_STAT_25K_015106", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.85, s=18.54, n=110, compute CI."}]}, "response": "95% CI ≈ [58.39, 65.31].", "meta": {"id": "WUA_STAT_25K_015107", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.21, μ=97.65, σ=22.28, compute z."}]}, "response": "z = -0.648.", "meta": {"id": "WUA_STAT_25K_015108", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.67, compute E and Var."}]}, "response": "E[X]=15.41; Var[X]=5.085.", "meta": {"id": "WUA_STAT_25K_015109", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.19, compute E and Var."}]}, "response": "E[X]=4.18; Var[X]=3.386.", "meta": {"id": "WUA_STAT_25K_015110", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.13, compute E and Var."}]}, "response": "E[X]=2.08; Var[X]=1.810.", "meta": {"id": "WUA_STAT_25K_015111", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.043, sens=0.89, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.154.", "meta": {"id": "WUA_STAT_25K_015112", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.8, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.273.", "meta": {"id": "WUA_STAT_25K_015113", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.67, s=9.3, n=182, compute CI."}]}, "response": "95% CI ≈ [173.32, 176.02].", "meta": {"id": "WUA_STAT_25K_015114", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.44, s=1.61, n=61, compute CI."}]}, "response": "95% CI ≈ [137.04, 137.84].", "meta": {"id": "WUA_STAT_25K_015115", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.015, sens=0.71, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.077.", "meta": {"id": "WUA_STAT_25K_015116", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.049, sens=0.71, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.119.", "meta": {"id": "WUA_STAT_25K_015117", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.45, s=18.57, n=167, compute CI."}]}, "response": "95% CI ≈ [67.63, 73.27].", "meta": {"id": "WUA_STAT_25K_015118", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.55, compute E and Var."}]}, "response": "E[X]=20.35; Var[X]=9.157.", "meta": {"id": "WUA_STAT_25K_015119", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.222, sens=0.92, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.814.", "meta": {"id": "WUA_STAT_25K_015120", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.21, μ=73.4, σ=11.05, compute z."}]}, "response": "z = -6.262.", "meta": {"id": "WUA_STAT_25K_015121", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.0, μ=56.6, σ=20.94, compute z."}]}, "response": "z = 1.404.", "meta": {"id": "WUA_STAT_25K_015122", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.78, compute E and Var."}]}, "response": "E[X]=7.02; Var[X]=1.544.", "meta": {"id": "WUA_STAT_25K_015123", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.66, compute E and Var."}]}, "response": "E[X]=16.50; Var[X]=5.610.", "meta": {"id": "WUA_STAT_25K_015124", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.03, s=31.18, n=141, compute CI."}]}, "response": "95% CI ≈ [101.88, 112.18].", "meta": {"id": "WUA_STAT_25K_015125", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.55, s=39.56, n=120, compute CI."}]}, "response": "95% CI ≈ [67.47, 81.63].", "meta": {"id": "WUA_STAT_25K_015126", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.32, compute E and Var."}]}, "response": "E[X]=11.84; Var[X]=8.051.", "meta": {"id": "WUA_STAT_25K_015127", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.23, μ=32.51, σ=19.91, compute z."}]}, "response": "z = -0.366.", "meta": {"id": "WUA_STAT_25K_015128", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.93, s=9.58, n=11, compute CI."}]}, "response": "95% CI ≈ [184.27, 195.59].", "meta": {"id": "WUA_STAT_25K_015129", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.191, sens=0.92, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.879.", "meta": {"id": "WUA_STAT_25K_015130", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.27, s=2.33, n=129, compute CI."}]}, "response": "95% CI ≈ [105.87, 106.67].", "meta": {"id": "WUA_STAT_25K_015131", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.12, compute E and Var."}]}, "response": "E[X]=1.80; Var[X]=1.584.", "meta": {"id": "WUA_STAT_25K_015132", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=134.02, s=8.43, n=106, compute CI."}]}, "response": "95% CI ≈ [132.42, 135.62].", "meta": {"id": "WUA_STAT_25K_015133", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.1, μ=18.34, σ=13.36, compute z."}]}, "response": "z = 4.922.", "meta": {"id": "WUA_STAT_25K_015134", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.1, s=38.03, n=190, compute CI."}]}, "response": "95% CI ≈ [115.69, 126.51].", "meta": {"id": "WUA_STAT_25K_015135", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.36, s=32.25, n=124, compute CI."}]}, "response": "95% CI ≈ [153.68, 165.04].", "meta": {"id": "WUA_STAT_25K_015136", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.58, compute E and Var."}]}, "response": "E[X]=15.66; Var[X]=6.577.", "meta": {"id": "WUA_STAT_25K_015137", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.06, μ=38.43, σ=16.44, compute z."}]}, "response": "z = -0.874.", "meta": {"id": "WUA_STAT_25K_015138", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.87, s=23.34, n=186, compute CI."}]}, "response": "95% CI ≈ [42.52, 49.22].", "meta": {"id": "WUA_STAT_25K_015139", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.56, compute E and Var."}]}, "response": "E[X]=16.24; Var[X]=7.146.", "meta": {"id": "WUA_STAT_25K_015140", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.69, compute E and Var."}]}, "response": "E[X]=27.60; Var[X]=8.556.", "meta": {"id": "WUA_STAT_25K_015141", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.65, compute E and Var."}]}, "response": "E[X]=23.40; Var[X]=8.190.", "meta": {"id": "WUA_STAT_25K_015142", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.69, compute E and Var."}]}, "response": "E[X]=24.84; Var[X]=7.700.", "meta": {"id": "WUA_STAT_25K_015143", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.46, compute E and Var."}]}, "response": "E[X]=10.12; Var[X]=5.465.", "meta": {"id": "WUA_STAT_25K_015144", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=105.69, s=31.49, n=87, compute CI."}]}, "response": "95% CI ≈ [99.07, 112.31].", "meta": {"id": "WUA_STAT_25K_015145", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.148, sens=0.96, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.544.", "meta": {"id": "WUA_STAT_25K_015146", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.89, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.432.", "meta": {"id": "WUA_STAT_25K_015147", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.96, s=32.1, n=138, compute CI."}]}, "response": "95% CI ≈ [148.60, 159.32].", "meta": {"id": "WUA_STAT_25K_015148", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.15, μ=47.21, σ=22.02, compute z."}]}, "response": "z = 0.906.", "meta": {"id": "WUA_STAT_25K_015149", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.73, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.219.", "meta": {"id": "WUA_STAT_25K_015150", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.68, μ=80.03, σ=8.2, compute z."}]}, "response": "z = 0.201.", "meta": {"id": "WUA_STAT_25K_015151", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.16, μ=51.89, σ=16.93, compute z."}]}, "response": "z = 2.556.", "meta": {"id": "WUA_STAT_25K_015152", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.29, compute E and Var."}]}, "response": "E[X]=5.51; Var[X]=3.912.", "meta": {"id": "WUA_STAT_25K_015153", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.4, compute E and Var."}]}, "response": "E[X]=12.80; Var[X]=7.680.", "meta": {"id": "WUA_STAT_25K_015154", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.3, s=11.55, n=55, compute CI."}]}, "response": "95% CI ≈ [107.25, 113.35].", "meta": {"id": "WUA_STAT_25K_015155", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=52.35, s=14.09, n=174, compute CI."}]}, "response": "95% CI ≈ [50.26, 54.44].", "meta": {"id": "WUA_STAT_25K_015156", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.87, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.866.", "meta": {"id": "WUA_STAT_25K_015157", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.104, sens=0.87, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.280.", "meta": {"id": "WUA_STAT_25K_015158", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.53, μ=32.22, σ=19.48, compute z."}]}, "response": "z = 3.045.", "meta": {"id": "WUA_STAT_25K_015159", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.17, compute E and Var."}]}, "response": "E[X]=6.29; Var[X]=5.221.", "meta": {"id": "WUA_STAT_25K_015160", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.57, compute E and Var."}]}, "response": "E[X]=22.23; Var[X]=9.559.", "meta": {"id": "WUA_STAT_25K_015161", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.71, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.533.", "meta": {"id": "WUA_STAT_25K_015162", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.053, sens=0.99, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.480.", "meta": {"id": "WUA_STAT_25K_015163", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.28, s=37.2, n=196, compute CI."}]}, "response": "95% CI ≈ [73.07, 83.49].", "meta": {"id": "WUA_STAT_25K_015164", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=189.85, s=25.74, n=150, compute CI."}]}, "response": "95% CI ≈ [185.73, 193.97].", "meta": {"id": "WUA_STAT_25K_015165", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.35, compute E and Var."}]}, "response": "E[X]=7.35; Var[X]=4.777.", "meta": {"id": "WUA_STAT_25K_015166", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.46, μ=95.3, σ=8.19, compute z."}]}, "response": "z = -11.458.", "meta": {"id": "WUA_STAT_25K_015167", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.53, compute E and Var."}]}, "response": "E[X]=7.95; Var[X]=3.736.", "meta": {"id": "WUA_STAT_25K_015168", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.23, s=7.4, n=66, compute CI."}]}, "response": "95% CI ≈ [185.44, 189.02].", "meta": {"id": "WUA_STAT_25K_015169", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.19, compute E and Var."}]}, "response": "E[X]=5.51; Var[X]=4.463.", "meta": {"id": "WUA_STAT_25K_015170", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.051, sens=0.87, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.609.", "meta": {"id": "WUA_STAT_25K_015171", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.11, s=36.74, n=29, compute CI."}]}, "response": "95% CI ≈ [20.74, 47.48].", "meta": {"id": "WUA_STAT_25K_015172", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.97, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.383.", "meta": {"id": "WUA_STAT_25K_015173", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.28, s=33.43, n=76, compute CI."}]}, "response": "95% CI ≈ [62.76, 77.80].", "meta": {"id": "WUA_STAT_25K_015174", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.85, μ=3.9, σ=14.05, compute z."}]}, "response": "z = 0.993.", "meta": {"id": "WUA_STAT_25K_015175", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.37, μ=23.74, σ=3.83, compute z."}]}, "response": "z = 14.264.", "meta": {"id": "WUA_STAT_25K_015176", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.6, μ=3.69, σ=14.45, compute z."}]}, "response": "z = -0.145.", "meta": {"id": "WUA_STAT_25K_015177", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.07, μ=56.52, σ=1.0, compute z."}]}, "response": "z = -44.450.", "meta": {"id": "WUA_STAT_25K_015178", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.32, compute E and Var."}]}, "response": "E[X]=2.56; Var[X]=1.741.", "meta": {"id": "WUA_STAT_25K_015179", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.98, s=28.44, n=77, compute CI."}]}, "response": "95% CI ≈ [174.63, 187.33].", "meta": {"id": "WUA_STAT_25K_015180", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.96, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.851.", "meta": {"id": "WUA_STAT_25K_015181", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.82, s=18.37, n=186, compute CI."}]}, "response": "95% CI ≈ [29.18, 34.46].", "meta": {"id": "WUA_STAT_25K_015182", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.75, s=38.93, n=67, compute CI."}]}, "response": "95% CI ≈ [36.43, 55.07].", "meta": {"id": "WUA_STAT_25K_015183", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.68, compute E and Var."}]}, "response": "E[X]=14.28; Var[X]=4.570.", "meta": {"id": "WUA_STAT_25K_015184", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.63, compute E and Var."}]}, "response": "E[X]=7.56; Var[X]=2.797.", "meta": {"id": "WUA_STAT_25K_015185", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.35, s=29.7, n=41, compute CI."}]}, "response": "95% CI ≈ [13.26, 31.44].", "meta": {"id": "WUA_STAT_25K_015186", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.65, compute E and Var."}]}, "response": "E[X]=22.10; Var[X]=7.735.", "meta": {"id": "WUA_STAT_25K_015187", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.46, s=39.56, n=116, compute CI."}]}, "response": "95% CI ≈ [47.26, 61.66].", "meta": {"id": "WUA_STAT_25K_015188", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.16, s=26.38, n=51, compute CI."}]}, "response": "95% CI ≈ [146.92, 161.40].", "meta": {"id": "WUA_STAT_25K_015189", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.96, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.243.", "meta": {"id": "WUA_STAT_25K_015190", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.79, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.558.", "meta": {"id": "WUA_STAT_25K_015191", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.27, μ=52.16, σ=17.2, compute z."}]}, "response": "z = -0.342.", "meta": {"id": "WUA_STAT_25K_015192", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=20.11, s=13.01, n=80, compute CI."}]}, "response": "95% CI ≈ [17.26, 22.96].", "meta": {"id": "WUA_STAT_25K_015193", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.55, s=38.39, n=57, compute CI."}]}, "response": "95% CI ≈ [146.58, 166.52].", "meta": {"id": "WUA_STAT_25K_015194", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.45, compute E and Var."}]}, "response": "E[X]=3.15; Var[X]=1.733.", "meta": {"id": "WUA_STAT_25K_015195", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=56.72, s=11.16, n=75, compute CI."}]}, "response": "95% CI ≈ [54.19, 59.25].", "meta": {"id": "WUA_STAT_25K_015196", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.08, μ=16.77, σ=16.19, compute z."}]}, "response": "z = 0.575.", "meta": {"id": "WUA_STAT_25K_015197", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.13, μ=51.68, σ=4.98, compute z."}]}, "response": "z = 7.319.", "meta": {"id": "WUA_STAT_25K_015198", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.68, μ=33.09, σ=7.64, compute z."}]}, "response": "z = 4.920.", "meta": {"id": "WUA_STAT_25K_015199", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.9, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.906.", "meta": {"id": "WUA_STAT_25K_015200", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.09, s=11.59, n=55, compute CI."}]}, "response": "95% CI ≈ [11.03, 17.15].", "meta": {"id": "WUA_STAT_25K_015201", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.71, μ=91.0, σ=15.89, compute z."}]}, "response": "z = -3.165.", "meta": {"id": "WUA_STAT_25K_015202", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.21, compute E and Var."}]}, "response": "E[X]=4.62; Var[X]=3.650.", "meta": {"id": "WUA_STAT_25K_015203", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=116.28, s=20.26, n=26, compute CI."}]}, "response": "95% CI ≈ [108.49, 124.07].", "meta": {"id": "WUA_STAT_25K_015204", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.61, compute E and Var."}]}, "response": "E[X]=21.35; Var[X]=8.326.", "meta": {"id": "WUA_STAT_25K_015205", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.66, compute E and Var."}]}, "response": "E[X]=17.82; Var[X]=6.059.", "meta": {"id": "WUA_STAT_25K_015206", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=26.86, s=14.87, n=28, compute CI."}]}, "response": "95% CI ≈ [21.35, 32.37].", "meta": {"id": "WUA_STAT_25K_015207", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.75, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.296.", "meta": {"id": "WUA_STAT_25K_015208", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.76, μ=51.21, σ=20.88, compute z."}]}, "response": "z = 0.026.", "meta": {"id": "WUA_STAT_25K_015209", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.86, μ=5.05, σ=17.51, compute z."}]}, "response": "z = 0.503.", "meta": {"id": "WUA_STAT_25K_015210", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.05, compute E and Var."}]}, "response": "E[X]=0.80; Var[X]=0.760.", "meta": {"id": "WUA_STAT_25K_015211", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.53, μ=34.46, σ=12.91, compute z."}]}, "response": "z = -0.382.", "meta": {"id": "WUA_STAT_25K_015212", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.17, μ=31.67, σ=2.5, compute z."}]}, "response": "z = -0.200.", "meta": {"id": "WUA_STAT_25K_015213", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.19, compute E and Var."}]}, "response": "E[X]=3.23; Var[X]=2.616.", "meta": {"id": "WUA_STAT_25K_015214", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.148, sens=0.77, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.656.", "meta": {"id": "WUA_STAT_25K_015215", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=170.76, s=34.13, n=153, compute CI."}]}, "response": "95% CI ≈ [165.35, 176.17].", "meta": {"id": "WUA_STAT_25K_015216", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.98, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.399.", "meta": {"id": "WUA_STAT_25K_015217", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.45, compute E and Var."}]}, "response": "E[X]=7.65; Var[X]=4.208.", "meta": {"id": "WUA_STAT_25K_015218", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.65, s=9.26, n=91, compute CI."}]}, "response": "95% CI ≈ [90.75, 94.55].", "meta": {"id": "WUA_STAT_25K_015219", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.21, compute E and Var."}]}, "response": "E[X]=5.88; Var[X]=4.645.", "meta": {"id": "WUA_STAT_25K_015220", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.63, compute E and Var."}]}, "response": "E[X]=22.05; Var[X]=8.159.", "meta": {"id": "WUA_STAT_25K_015221", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.26, compute E and Var."}]}, "response": "E[X]=4.42; Var[X]=3.271.", "meta": {"id": "WUA_STAT_25K_015222", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.87, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.320.", "meta": {"id": "WUA_STAT_25K_015223", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.19, s=13.83, n=149, compute CI."}]}, "response": "95% CI ≈ [9.97, 14.41].", "meta": {"id": "WUA_STAT_25K_015224", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.28, compute E and Var."}]}, "response": "E[X]=10.92; Var[X]=7.862.", "meta": {"id": "WUA_STAT_25K_015225", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.62, μ=81.64, σ=15.3, compute z."}]}, "response": "z = -0.393.", "meta": {"id": "WUA_STAT_25K_015226", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.21, compute E and Var."}]}, "response": "E[X]=2.73; Var[X]=2.157.", "meta": {"id": "WUA_STAT_25K_015227", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.55, compute E and Var."}]}, "response": "E[X]=8.80; Var[X]=3.960.", "meta": {"id": "WUA_STAT_25K_015228", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.71, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.125.", "meta": {"id": "WUA_STAT_25K_015229", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.89, s=9.83, n=12, compute CI."}]}, "response": "95% CI ≈ [113.33, 124.45].", "meta": {"id": "WUA_STAT_25K_015230", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.82, μ=59.56, σ=3.22, compute z."}]}, "response": "z = -17.932.", "meta": {"id": "WUA_STAT_25K_015231", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.78, compute E and Var."}]}, "response": "E[X]=24.18; Var[X]=5.320.", "meta": {"id": "WUA_STAT_25K_015232", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.224, sens=0.73, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.421.", "meta": {"id": "WUA_STAT_25K_015233", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.83, s=20.42, n=16, compute CI."}]}, "response": "95% CI ≈ [55.82, 75.84].", "meta": {"id": "WUA_STAT_25K_015234", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.61, s=17.17, n=100, compute CI."}]}, "response": "95% CI ≈ [122.24, 128.98].", "meta": {"id": "WUA_STAT_25K_015235", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.73, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.902.", "meta": {"id": "WUA_STAT_25K_015236", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.53, s=19.33, n=61, compute CI."}]}, "response": "95% CI ≈ [115.68, 125.38].", "meta": {"id": "WUA_STAT_25K_015237", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.43, compute E and Var."}]}, "response": "E[X]=4.73; Var[X]=2.696.", "meta": {"id": "WUA_STAT_25K_015238", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.46, μ=34.99, σ=1.78, compute z."}]}, "response": "z = 30.039.", "meta": {"id": "WUA_STAT_25K_015239", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.49, compute E and Var."}]}, "response": "E[X]=11.27; Var[X]=5.748.", "meta": {"id": "WUA_STAT_25K_015240", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.37, s=10.8, n=33, compute CI."}]}, "response": "95% CI ≈ [26.69, 34.05].", "meta": {"id": "WUA_STAT_25K_015241", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.86, μ=4.33, σ=14.46, compute z."}]}, "response": "z = 0.659.", "meta": {"id": "WUA_STAT_25K_015242", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=191.9, s=28.98, n=189, compute CI."}]}, "response": "95% CI ≈ [187.77, 196.03].", "meta": {"id": "WUA_STAT_25K_015243", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.17, μ=89.28, σ=4.3, compute z."}]}, "response": "z = -20.258.", "meta": {"id": "WUA_STAT_25K_015244", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.049, sens=0.77, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.173.", "meta": {"id": "WUA_STAT_25K_015245", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.95, μ=13.11, σ=1.57, compute z."}]}, "response": "z = 26.013.", "meta": {"id": "WUA_STAT_25K_015246", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.56, μ=63.49, σ=3.03, compute z."}]}, "response": "z = 10.914.", "meta": {"id": "WUA_STAT_25K_015247", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.71, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.208.", "meta": {"id": "WUA_STAT_25K_015248", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.44, s=16.1, n=123, compute CI."}]}, "response": "95% CI ≈ [80.59, 86.29].", "meta": {"id": "WUA_STAT_25K_015249", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.34, μ=72.21, σ=21.42, compute z."}]}, "response": "z = 0.426.", "meta": {"id": "WUA_STAT_25K_015250", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.46, compute E and Var."}]}, "response": "E[X]=4.14; Var[X]=2.236.", "meta": {"id": "WUA_STAT_25K_015251", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.028, sens=0.94, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.162.", "meta": {"id": "WUA_STAT_25K_015252", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.09, compute E and Var."}]}, "response": "E[X]=2.61; Var[X]=2.375.", "meta": {"id": "WUA_STAT_25K_015253", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.79, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.277.", "meta": {"id": "WUA_STAT_25K_015254", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.13, compute E and Var."}]}, "response": "E[X]=3.77; Var[X]=3.280.", "meta": {"id": "WUA_STAT_25K_015255", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.03, μ=45.47, σ=16.32, compute z."}]}, "response": "z = -0.456.", "meta": {"id": "WUA_STAT_25K_015256", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.5, s=16.19, n=79, compute CI."}]}, "response": "95% CI ≈ [149.93, 157.07].", "meta": {"id": "WUA_STAT_25K_015257", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.1, compute E and Var."}]}, "response": "E[X]=2.00; Var[X]=1.800.", "meta": {"id": "WUA_STAT_25K_015258", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.02, s=3.02, n=104, compute CI."}]}, "response": "95% CI ≈ [131.44, 132.60].", "meta": {"id": "WUA_STAT_25K_015259", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.14, sens=0.84, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.354.", "meta": {"id": "WUA_STAT_25K_015260", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.69, compute E and Var."}]}, "response": "E[X]=21.39; Var[X]=6.631.", "meta": {"id": "WUA_STAT_25K_015261", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.4, μ=23.49, σ=16.79, compute z."}]}, "response": "z = -0.541.", "meta": {"id": "WUA_STAT_25K_015262", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.46, μ=59.61, σ=15.19, compute z."}]}, "response": "z = 0.385.", "meta": {"id": "WUA_STAT_25K_015263", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.9, s=12.52, n=91, compute CI."}]}, "response": "95% CI ≈ [184.33, 189.47].", "meta": {"id": "WUA_STAT_25K_015264", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.64, compute E and Var."}]}, "response": "E[X]=21.76; Var[X]=7.834.", "meta": {"id": "WUA_STAT_25K_015265", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=95.11, μ=73.1, σ=17.37, compute z."}]}, "response": "z = 1.267.", "meta": {"id": "WUA_STAT_25K_015266", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.79, μ=64.9, σ=4.36, compute z."}]}, "response": "z = -4.154.", "meta": {"id": "WUA_STAT_25K_015267", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.09, sens=0.81, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.534.", "meta": {"id": "WUA_STAT_25K_015268", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.067, sens=0.79, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.205.", "meta": {"id": "WUA_STAT_25K_015269", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.79, μ=81.73, σ=11.46, compute z."}]}, "response": "z = -4.009.", "meta": {"id": "WUA_STAT_25K_015270", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.58, μ=52.47, σ=13.95, compute z."}]}, "response": "z = 2.732.", "meta": {"id": "WUA_STAT_25K_015271", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.57, compute E and Var."}]}, "response": "E[X]=21.09; Var[X]=9.069.", "meta": {"id": "WUA_STAT_25K_015272", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.45, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=3.960.", "meta": {"id": "WUA_STAT_25K_015273", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.68, s=19.44, n=28, compute CI."}]}, "response": "95% CI ≈ [51.48, 65.88].", "meta": {"id": "WUA_STAT_25K_015274", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.84, μ=45.04, σ=3.91, compute z."}]}, "response": "z = 4.297.", "meta": {"id": "WUA_STAT_25K_015275", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.046, sens=0.87, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.677.", "meta": {"id": "WUA_STAT_25K_015276", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.03, sens=0.81, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.102.", "meta": {"id": "WUA_STAT_25K_015277", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.74, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.444.", "meta": {"id": "WUA_STAT_25K_015278", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=10.69, s=26.8, n=164, compute CI."}]}, "response": "95% CI ≈ [6.59, 14.79].", "meta": {"id": "WUA_STAT_25K_015279", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.74, μ=34.19, σ=1.12, compute z."}]}, "response": "z = -19.152.", "meta": {"id": "WUA_STAT_25K_015280", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.65, compute E and Var."}]}, "response": "E[X]=9.75; Var[X]=3.412.", "meta": {"id": "WUA_STAT_25K_015281", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.44, μ=72.36, σ=1.82, compute z."}]}, "response": "z = -15.890.", "meta": {"id": "WUA_STAT_25K_015282", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.57, μ=5.17, σ=2.05, compute z."}]}, "response": "z = -0.293.", "meta": {"id": "WUA_STAT_25K_015283", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.107, sens=0.9, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.473.", "meta": {"id": "WUA_STAT_25K_015284", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=158.48, s=23.75, n=129, compute CI."}]}, "response": "95% CI ≈ [154.38, 162.58].", "meta": {"id": "WUA_STAT_25K_015285", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.51, s=32.86, n=153, compute CI."}]}, "response": "95% CI ≈ [31.30, 41.72].", "meta": {"id": "WUA_STAT_25K_015286", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.18, compute E and Var."}]}, "response": "E[X]=4.86; Var[X]=3.985.", "meta": {"id": "WUA_STAT_25K_015287", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.13, sens=0.85, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.459.", "meta": {"id": "WUA_STAT_25K_015288", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.232, sens=0.95, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.534.", "meta": {"id": "WUA_STAT_25K_015289", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.9, s=22.5, n=66, compute CI."}]}, "response": "95% CI ≈ [114.47, 125.33].", "meta": {"id": "WUA_STAT_25K_015290", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.117, sens=0.96, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.645.", "meta": {"id": "WUA_STAT_25K_015291", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.83, μ=97.14, σ=22.8, compute z."}]}, "response": "z = -3.259.", "meta": {"id": "WUA_STAT_25K_015292", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.62, s=20.17, n=73, compute CI."}]}, "response": "95% CI ≈ [179.99, 189.25].", "meta": {"id": "WUA_STAT_25K_015293", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.72, s=35.51, n=134, compute CI."}]}, "response": "95% CI ≈ [68.71, 80.73].", "meta": {"id": "WUA_STAT_25K_015294", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.26, compute E and Var."}]}, "response": "E[X]=9.62; Var[X]=7.119.", "meta": {"id": "WUA_STAT_25K_015295", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.7, μ=80.96, σ=4.16, compute z."}]}, "response": "z = -14.245.", "meta": {"id": "WUA_STAT_25K_015296", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.1, μ=26.85, σ=7.69, compute z."}]}, "response": "z = 3.153.", "meta": {"id": "WUA_STAT_25K_015297", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.28, μ=38.65, σ=17.55, compute z."}]}, "response": "z = 1.631.", "meta": {"id": "WUA_STAT_25K_015298", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.11, μ=67.1, σ=23.84, compute z."}]}, "response": "z = -0.125.", "meta": {"id": "WUA_STAT_25K_015299", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.38, μ=16.25, σ=5.0, compute z."}]}, "response": "z = 12.626.", "meta": {"id": "WUA_STAT_25K_015300", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.88, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.429.", "meta": {"id": "WUA_STAT_25K_015301", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.36, μ=99.73, σ=24.79, compute z."}]}, "response": "z = -1.790.", "meta": {"id": "WUA_STAT_25K_015302", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.22, μ=31.82, σ=22.41, compute z."}]}, "response": "z = 0.553.", "meta": {"id": "WUA_STAT_25K_015303", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.64, μ=5.24, σ=4.31, compute z."}]}, "response": "z = 2.645.", "meta": {"id": "WUA_STAT_25K_015304", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.83, s=15.51, n=13, compute CI."}]}, "response": "95% CI ≈ [55.40, 72.26].", "meta": {"id": "WUA_STAT_25K_015305", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.38, compute E and Var."}]}, "response": "E[X]=5.70; Var[X]=3.534.", "meta": {"id": "WUA_STAT_25K_015306", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.91, μ=61.66, σ=9.01, compute z."}]}, "response": "z = 1.027.", "meta": {"id": "WUA_STAT_25K_015307", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.71, μ=97.41, σ=9.56, compute z."}]}, "response": "z = -5.931.", "meta": {"id": "WUA_STAT_25K_015308", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.043, sens=0.83, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.253.", "meta": {"id": "WUA_STAT_25K_015309", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.67, s=20.23, n=66, compute CI."}]}, "response": "95% CI ≈ [7.79, 17.55].", "meta": {"id": "WUA_STAT_25K_015310", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.187, sens=0.73, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.604.", "meta": {"id": "WUA_STAT_25K_015311", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.17, sens=0.95, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.907.", "meta": {"id": "WUA_STAT_25K_015312", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.74, s=16.78, n=194, compute CI."}]}, "response": "95% CI ≈ [28.38, 33.10].", "meta": {"id": "WUA_STAT_25K_015313", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.77, μ=88.96, σ=23.79, compute z."}]}, "response": "z = -2.698.", "meta": {"id": "WUA_STAT_25K_015314", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.08, compute E and Var."}]}, "response": "E[X]=2.64; Var[X]=2.429.", "meta": {"id": "WUA_STAT_25K_015315", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.03, μ=32.47, σ=7.53, compute z."}]}, "response": "z = 6.316.", "meta": {"id": "WUA_STAT_25K_015316", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.17, μ=89.14, σ=10.45, compute z."}]}, "response": "z = -1.145.", "meta": {"id": "WUA_STAT_25K_015317", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.16, μ=86.85, σ=14.72, compute z."}]}, "response": "z = -4.327.", "meta": {"id": "WUA_STAT_25K_015318", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.45, s=33.71, n=93, compute CI."}]}, "response": "95% CI ≈ [70.60, 84.30].", "meta": {"id": "WUA_STAT_25K_015319", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=91.88, s=12.14, n=165, compute CI."}]}, "response": "95% CI ≈ [90.03, 93.73].", "meta": {"id": "WUA_STAT_25K_015320", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.41, s=25.58, n=141, compute CI."}]}, "response": "95% CI ≈ [11.19, 19.63].", "meta": {"id": "WUA_STAT_25K_015321", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.98, μ=38.98, σ=12.22, compute z."}]}, "response": "z = 1.309.", "meta": {"id": "WUA_STAT_25K_015322", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.014, sens=0.82, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.127.", "meta": {"id": "WUA_STAT_25K_015323", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.57, μ=54.22, σ=1.35, compute z."}]}, "response": "z = 15.815.", "meta": {"id": "WUA_STAT_25K_015324", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.73, μ=35.26, σ=8.72, compute z."}]}, "response": "z = -2.240.", "meta": {"id": "WUA_STAT_25K_015325", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.14, compute E and Var."}]}, "response": "E[X]=0.70; Var[X]=0.602.", "meta": {"id": "WUA_STAT_25K_015326", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.58, compute E and Var."}]}, "response": "E[X]=13.34; Var[X]=5.603.", "meta": {"id": "WUA_STAT_25K_015327", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.88, μ=18.75, σ=15.18, compute z."}]}, "response": "z = 0.536.", "meta": {"id": "WUA_STAT_25K_015328", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.05, compute E and Var."}]}, "response": "E[X]=1.35; Var[X]=1.282.", "meta": {"id": "WUA_STAT_25K_015329", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=120.49, s=29.08, n=55, compute CI."}]}, "response": "95% CI ≈ [112.80, 128.18].", "meta": {"id": "WUA_STAT_25K_015330", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.08, compute E and Var."}]}, "response": "E[X]=1.04; Var[X]=0.957.", "meta": {"id": "WUA_STAT_25K_015331", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.62, compute E and Var."}]}, "response": "E[X]=22.32; Var[X]=8.482.", "meta": {"id": "WUA_STAT_25K_015332", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=19.81, s=11.44, n=15, compute CI."}]}, "response": "95% CI ≈ [14.02, 25.60].", "meta": {"id": "WUA_STAT_25K_015333", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.83, s=34.2, n=97, compute CI."}]}, "response": "95% CI ≈ [7.02, 20.64].", "meta": {"id": "WUA_STAT_25K_015334", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.83, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.642.", "meta": {"id": "WUA_STAT_25K_015335", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.01, sens=0.99, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.036.", "meta": {"id": "WUA_STAT_25K_015336", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.49, μ=28.81, σ=10.79, compute z."}]}, "response": "z = 5.809.", "meta": {"id": "WUA_STAT_25K_015337", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.18, μ=46.46, σ=19.64, compute z."}]}, "response": "z = -0.422.", "meta": {"id": "WUA_STAT_25K_015338", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.63, compute E and Var."}]}, "response": "E[X]=17.64; Var[X]=6.527.", "meta": {"id": "WUA_STAT_25K_015339", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.141, sens=0.93, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.836.", "meta": {"id": "WUA_STAT_25K_015340", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.85, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.275.", "meta": {"id": "WUA_STAT_25K_015341", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.69, compute E and Var."}]}, "response": "E[X]=10.35; Var[X]=3.209.", "meta": {"id": "WUA_STAT_25K_015342", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.57, compute E and Var."}]}, "response": "E[X]=5.13; Var[X]=2.206.", "meta": {"id": "WUA_STAT_25K_015343", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.21, compute E and Var."}]}, "response": "E[X]=2.10; Var[X]=1.659.", "meta": {"id": "WUA_STAT_25K_015344", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.72, compute E and Var."}]}, "response": "E[X]=12.24; Var[X]=3.427.", "meta": {"id": "WUA_STAT_25K_015345", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.33, μ=0.92, σ=13.06, compute z."}]}, "response": "z = 3.630.", "meta": {"id": "WUA_STAT_25K_015346", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=106.63, s=20.38, n=15, compute CI."}]}, "response": "95% CI ≈ [96.32, 116.94].", "meta": {"id": "WUA_STAT_25K_015347", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.47, compute E and Var."}]}, "response": "E[X]=18.80; Var[X]=9.964.", "meta": {"id": "WUA_STAT_25K_015348", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.61, μ=3.47, σ=13.69, compute z."}]}, "response": "z = -0.136.", "meta": {"id": "WUA_STAT_25K_015349", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.42, s=11.09, n=62, compute CI."}]}, "response": "95% CI ≈ [135.66, 141.18].", "meta": {"id": "WUA_STAT_25K_015350", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.59, compute E and Var."}]}, "response": "E[X]=6.49; Var[X]=2.661.", "meta": {"id": "WUA_STAT_25K_015351", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=47.15, s=6.12, n=82, compute CI."}]}, "response": "95% CI ≈ [45.83, 48.47].", "meta": {"id": "WUA_STAT_25K_015352", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.26, compute E and Var."}]}, "response": "E[X]=10.40; Var[X]=7.696.", "meta": {"id": "WUA_STAT_25K_015353", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.3, compute E and Var."}]}, "response": "E[X]=11.10; Var[X]=7.770.", "meta": {"id": "WUA_STAT_25K_015354", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.86, s=33.3, n=115, compute CI."}]}, "response": "95% CI ≈ [187.77, 199.95].", "meta": {"id": "WUA_STAT_25K_015355", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.43, s=14.67, n=105, compute CI."}]}, "response": "95% CI ≈ [195.62, 201.24].", "meta": {"id": "WUA_STAT_25K_015356", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.75, compute E and Var."}]}, "response": "E[X]=13.50; Var[X]=3.375.", "meta": {"id": "WUA_STAT_25K_015357", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.39, compute E and Var."}]}, "response": "E[X]=11.70; Var[X]=7.137.", "meta": {"id": "WUA_STAT_25K_015358", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.24, s=18.02, n=114, compute CI."}]}, "response": "95% CI ≈ [10.93, 17.55].", "meta": {"id": "WUA_STAT_25K_015359", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.32, s=19.59, n=36, compute CI."}]}, "response": "95% CI ≈ [58.92, 71.72].", "meta": {"id": "WUA_STAT_25K_015360", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.06, compute E and Var."}]}, "response": "E[X]=2.34; Var[X]=2.200.", "meta": {"id": "WUA_STAT_25K_015361", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.25, μ=53.65, σ=7.51, compute z."}]}, "response": "z = -0.053.", "meta": {"id": "WUA_STAT_25K_015362", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.57, μ=53.46, σ=19.29, compute z."}]}, "response": "z = 1.094.", "meta": {"id": "WUA_STAT_25K_015363", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.56, compute E and Var."}]}, "response": "E[X]=17.92; Var[X]=7.885.", "meta": {"id": "WUA_STAT_25K_015364", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.71, s=2.09, n=34, compute CI."}]}, "response": "95% CI ≈ [33.01, 34.41].", "meta": {"id": "WUA_STAT_25K_015365", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.05, compute E and Var."}]}, "response": "E[X]=2.00; Var[X]=1.900.", "meta": {"id": "WUA_STAT_25K_015366", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.4, μ=23.96, σ=13.41, compute z."}]}, "response": "z = 5.551.", "meta": {"id": "WUA_STAT_25K_015367", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.63, compute E and Var."}]}, "response": "E[X]=18.90; Var[X]=6.993.", "meta": {"id": "WUA_STAT_25K_015368", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.176, sens=0.97, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.597.", "meta": {"id": "WUA_STAT_25K_015369", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.203, sens=0.99, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.546.", "meta": {"id": "WUA_STAT_25K_015370", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.19, μ=49.05, σ=6.74, compute z."}]}, "response": "z = -1.315.", "meta": {"id": "WUA_STAT_25K_015371", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.048, sens=0.93, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.169.", "meta": {"id": "WUA_STAT_25K_015372", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.17, μ=10.73, σ=22.9, compute z."}]}, "response": "z = 2.596.", "meta": {"id": "WUA_STAT_25K_015373", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.06, compute E and Var."}]}, "response": "E[X]=2.16; Var[X]=2.030.", "meta": {"id": "WUA_STAT_25K_015374", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.74, compute E and Var."}]}, "response": "E[X]=16.28; Var[X]=4.233.", "meta": {"id": "WUA_STAT_25K_015375", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.01, μ=32.13, σ=17.43, compute z."}]}, "response": "z = 2.173.", "meta": {"id": "WUA_STAT_25K_015376", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.23, compute E and Var."}]}, "response": "E[X]=2.07; Var[X]=1.594.", "meta": {"id": "WUA_STAT_25K_015377", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.65, μ=93.84, σ=4.15, compute z."}]}, "response": "z = -11.130.", "meta": {"id": "WUA_STAT_25K_015378", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.45, compute E and Var."}]}, "response": "E[X]=16.20; Var[X]=8.910.", "meta": {"id": "WUA_STAT_25K_015379", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.22, compute E and Var."}]}, "response": "E[X]=2.20; Var[X]=1.716.", "meta": {"id": "WUA_STAT_25K_015380", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.38, μ=34.13, σ=21.14, compute z."}]}, "response": "z = 2.850.", "meta": {"id": "WUA_STAT_25K_015381", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.73, compute E and Var."}]}, "response": "E[X]=18.25; Var[X]=4.928.", "meta": {"id": "WUA_STAT_25K_015382", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.78, μ=8.51, σ=9.67, compute z."}]}, "response": "z = 3.544.", "meta": {"id": "WUA_STAT_25K_015383", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.94, μ=27.8, σ=2.96, compute z."}]}, "response": "z = -1.980.", "meta": {"id": "WUA_STAT_25K_015384", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.08, s=15.39, n=143, compute CI."}]}, "response": "95% CI ≈ [122.56, 127.60].", "meta": {"id": "WUA_STAT_25K_015385", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.08, s=10.8, n=174, compute CI."}]}, "response": "95% CI ≈ [47.48, 50.68].", "meta": {"id": "WUA_STAT_25K_015386", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.41, μ=30.88, σ=11.07, compute z."}]}, "response": "z = -0.765.", "meta": {"id": "WUA_STAT_25K_015387", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.094, sens=0.94, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.351.", "meta": {"id": "WUA_STAT_25K_015388", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.32, compute E and Var."}]}, "response": "E[X]=9.28; Var[X]=6.310.", "meta": {"id": "WUA_STAT_25K_015389", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.63, compute E and Var."}]}, "response": "E[X]=8.19; Var[X]=3.030.", "meta": {"id": "WUA_STAT_25K_015390", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.61, compute E and Var."}]}, "response": "E[X]=4.88; Var[X]=1.903.", "meta": {"id": "WUA_STAT_25K_015391", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.17, μ=84.15, σ=10.76, compute z."}]}, "response": "z = -6.411.", "meta": {"id": "WUA_STAT_25K_015392", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.75, μ=66.84, σ=20.1, compute z."}]}, "response": "z = -1.099.", "meta": {"id": "WUA_STAT_25K_015393", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.35, μ=43.13, σ=6.8, compute z."}]}, "response": "z = -5.262.", "meta": {"id": "WUA_STAT_25K_015394", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.35, μ=38.21, σ=23.7, compute z."}]}, "response": "z = 0.343.", "meta": {"id": "WUA_STAT_25K_015395", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.168, sens=0.78, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.512.", "meta": {"id": "WUA_STAT_25K_015396", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.23, μ=20.25, σ=21.85, compute z."}]}, "response": "z = -0.687.", "meta": {"id": "WUA_STAT_25K_015397", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.23, μ=96.85, σ=23.7, compute z."}]}, "response": "z = -1.672.", "meta": {"id": "WUA_STAT_25K_015398", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.104, sens=0.73, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.629.", "meta": {"id": "WUA_STAT_25K_015399", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=153.34, s=14.07, n=61, compute CI."}]}, "response": "95% CI ≈ [149.81, 156.87].", "meta": {"id": "WUA_STAT_25K_015400", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.86, μ=27.62, σ=17.17, compute z."}]}, "response": "z = 1.761.", "meta": {"id": "WUA_STAT_25K_015401", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.87, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.479.", "meta": {"id": "WUA_STAT_25K_015402", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.191, sens=0.85, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.741.", "meta": {"id": "WUA_STAT_25K_015403", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.85, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.490.", "meta": {"id": "WUA_STAT_25K_015404", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.24, s=37.23, n=175, compute CI."}]}, "response": "95% CI ≈ [55.72, 66.76].", "meta": {"id": "WUA_STAT_25K_015405", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.211, sens=0.72, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.456.", "meta": {"id": "WUA_STAT_25K_015406", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.39, μ=3.53, σ=24.75, compute z."}]}, "response": "z = 3.509.", "meta": {"id": "WUA_STAT_25K_015407", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.81, s=12.13, n=114, compute CI."}]}, "response": "95% CI ≈ [188.58, 193.04].", "meta": {"id": "WUA_STAT_25K_015408", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.92, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.880.", "meta": {"id": "WUA_STAT_25K_015409", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.07, compute E and Var."}]}, "response": "E[X]=2.24; Var[X]=2.083.", "meta": {"id": "WUA_STAT_25K_015410", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.237, sens=0.88, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.566.", "meta": {"id": "WUA_STAT_25K_015411", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.19, μ=37.87, σ=16.03, compute z."}]}, "response": "z = 2.952.", "meta": {"id": "WUA_STAT_25K_015412", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=116.05, s=12.47, n=197, compute CI."}]}, "response": "95% CI ≈ [114.31, 117.79].", "meta": {"id": "WUA_STAT_25K_015413", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.92, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.421.", "meta": {"id": "WUA_STAT_25K_015414", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.71, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.036.", "meta": {"id": "WUA_STAT_25K_015415", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.95, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.859.", "meta": {"id": "WUA_STAT_25K_015416", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.51, s=39.19, n=49, compute CI."}]}, "response": "95% CI ≈ [48.54, 70.48].", "meta": {"id": "WUA_STAT_25K_015417", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.47, compute E and Var."}]}, "response": "E[X]=16.92; Var[X]=8.968.", "meta": {"id": "WUA_STAT_25K_015418", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.27, compute E and Var."}]}, "response": "E[X]=2.43; Var[X]=1.774.", "meta": {"id": "WUA_STAT_25K_015419", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.28, μ=14.59, σ=2.82, compute z."}]}, "response": "z = 16.911.", "meta": {"id": "WUA_STAT_25K_015420", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.193, sens=0.9, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.574.", "meta": {"id": "WUA_STAT_25K_015421", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.48, compute E and Var."}]}, "response": "E[X]=12.00; Var[X]=6.240.", "meta": {"id": "WUA_STAT_25K_015422", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.27, μ=34.06, σ=9.2, compute z."}]}, "response": "z = -2.368.", "meta": {"id": "WUA_STAT_25K_015423", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.29, s=22.6, n=75, compute CI."}]}, "response": "95% CI ≈ [183.18, 193.40].", "meta": {"id": "WUA_STAT_25K_015424", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.6, compute E and Var."}]}, "response": "E[X]=13.80; Var[X]=5.520.", "meta": {"id": "WUA_STAT_25K_015425", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.03, s=38.82, n=20, compute CI."}]}, "response": "95% CI ≈ [1.02, 35.04].", "meta": {"id": "WUA_STAT_25K_015426", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.45, compute E and Var."}]}, "response": "E[X]=11.70; Var[X]=6.435.", "meta": {"id": "WUA_STAT_25K_015427", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.81, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.530.", "meta": {"id": "WUA_STAT_25K_015428", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.1, compute E and Var."}]}, "response": "E[X]=0.50; Var[X]=0.450.", "meta": {"id": "WUA_STAT_25K_015429", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.25, sens=0.72, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.571.", "meta": {"id": "WUA_STAT_25K_015430", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=77.59, μ=2.4, σ=4.58, compute z."}]}, "response": "z = 16.417.", "meta": {"id": "WUA_STAT_25K_015431", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.146, sens=0.79, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.403.", "meta": {"id": "WUA_STAT_25K_015432", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.016, sens=0.9, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.268.", "meta": {"id": "WUA_STAT_25K_015433", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.24, s=6.66, n=73, compute CI."}]}, "response": "95% CI ≈ [63.71, 66.77].", "meta": {"id": "WUA_STAT_25K_015434", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.18, sens=0.85, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.538.", "meta": {"id": "WUA_STAT_25K_015435", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.53, compute E and Var."}]}, "response": "E[X]=13.78; Var[X]=6.477.", "meta": {"id": "WUA_STAT_25K_015436", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.1, s=17.35, n=152, compute CI."}]}, "response": "95% CI ≈ [120.34, 125.86].", "meta": {"id": "WUA_STAT_25K_015437", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.55, compute E and Var."}]}, "response": "E[X]=16.50; Var[X]=7.425.", "meta": {"id": "WUA_STAT_25K_015438", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.044, sens=0.83, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.116.", "meta": {"id": "WUA_STAT_25K_015439", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.162, sens=0.83, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.364.", "meta": {"id": "WUA_STAT_25K_015440", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.5, compute E and Var."}]}, "response": "E[X]=11.00; Var[X]=5.500.", "meta": {"id": "WUA_STAT_25K_015441", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.12, s=15.33, n=194, compute CI."}]}, "response": "95% CI ≈ [95.96, 100.28].", "meta": {"id": "WUA_STAT_25K_015442", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.62, μ=39.18, σ=5.57, compute z."}]}, "response": "z = 6.183.", "meta": {"id": "WUA_STAT_25K_015443", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.76, compute E and Var."}]}, "response": "E[X]=17.48; Var[X]=4.195.", "meta": {"id": "WUA_STAT_25K_015444", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.09, compute E and Var."}]}, "response": "E[X]=2.16; Var[X]=1.966.", "meta": {"id": "WUA_STAT_25K_015445", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.094, sens=0.88, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.349.", "meta": {"id": "WUA_STAT_25K_015446", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.31, compute E and Var."}]}, "response": "E[X]=2.48; Var[X]=1.711.", "meta": {"id": "WUA_STAT_25K_015447", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.86, s=34.74, n=144, compute CI."}]}, "response": "95% CI ≈ [12.19, 23.53].", "meta": {"id": "WUA_STAT_25K_015448", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.52, compute E and Var."}]}, "response": "E[X]=12.48; Var[X]=5.990.", "meta": {"id": "WUA_STAT_25K_015449", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.94, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.509.", "meta": {"id": "WUA_STAT_25K_015450", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.031, sens=0.98, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.108.", "meta": {"id": "WUA_STAT_25K_015451", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=171.96, s=5.23, n=28, compute CI."}]}, "response": "95% CI ≈ [170.02, 173.90].", "meta": {"id": "WUA_STAT_25K_015452", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.06, μ=89.83, σ=9.17, compute z."}]}, "response": "z = -9.462.", "meta": {"id": "WUA_STAT_25K_015453", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=73.85, s=33.02, n=92, compute CI."}]}, "response": "95% CI ≈ [67.10, 80.60].", "meta": {"id": "WUA_STAT_25K_015454", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.3, compute E and Var."}]}, "response": "E[X]=5.10; Var[X]=3.570.", "meta": {"id": "WUA_STAT_25K_015455", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.57, s=36.78, n=13, compute CI."}]}, "response": "95% CI ≈ [156.58, 196.56].", "meta": {"id": "WUA_STAT_25K_015456", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.1, s=19.86, n=40, compute CI."}]}, "response": "95% CI ≈ [14.95, 27.25].", "meta": {"id": "WUA_STAT_25K_015457", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.5, μ=10.77, σ=11.34, compute z."}]}, "response": "z = 5.444.", "meta": {"id": "WUA_STAT_25K_015458", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.67, μ=46.41, σ=1.94, compute z."}]}, "response": "z = -8.113.", "meta": {"id": "WUA_STAT_25K_015459", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=28.03, s=29.17, n=59, compute CI."}]}, "response": "95% CI ≈ [20.59, 35.47].", "meta": {"id": "WUA_STAT_25K_015460", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.19, compute E and Var."}]}, "response": "E[X]=5.70; Var[X]=4.617.", "meta": {"id": "WUA_STAT_25K_015461", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.65, s=34.05, n=11, compute CI."}]}, "response": "95% CI ≈ [81.53, 121.77].", "meta": {"id": "WUA_STAT_25K_015462", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.22, μ=79.66, σ=6.74, compute z."}]}, "response": "z = -6.742.", "meta": {"id": "WUA_STAT_25K_015463", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.43, μ=86.81, σ=2.31, compute z."}]}, "response": "z = -7.957.", "meta": {"id": "WUA_STAT_25K_015464", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.88, μ=96.81, σ=1.77, compute z."}]}, "response": "z = -25.384.", "meta": {"id": "WUA_STAT_25K_015465", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.37, compute E and Var."}]}, "response": "E[X]=2.59; Var[X]=1.632.", "meta": {"id": "WUA_STAT_25K_015466", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.81, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.494.", "meta": {"id": "WUA_STAT_25K_015467", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.98, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.841.", "meta": {"id": "WUA_STAT_25K_015468", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.84, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.064.", "meta": {"id": "WUA_STAT_25K_015469", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.26, compute E and Var."}]}, "response": "E[X]=2.08; Var[X]=1.539.", "meta": {"id": "WUA_STAT_25K_015470", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.99, μ=33.73, σ=22.37, compute z."}]}, "response": "z = -0.435.", "meta": {"id": "WUA_STAT_25K_015471", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.38, μ=34.83, σ=23.91, compute z."}]}, "response": "z = 0.734.", "meta": {"id": "WUA_STAT_25K_015472", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.99, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.903.", "meta": {"id": "WUA_STAT_25K_015473", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.96, s=28.95, n=180, compute CI."}]}, "response": "95% CI ≈ [55.73, 64.19].", "meta": {"id": "WUA_STAT_25K_015474", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.193, sens=0.93, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.614.", "meta": {"id": "WUA_STAT_25K_015475", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.74, s=38.01, n=112, compute CI."}]}, "response": "95% CI ≈ [128.70, 142.78].", "meta": {"id": "WUA_STAT_25K_015476", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.095, sens=0.79, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.222.", "meta": {"id": "WUA_STAT_25K_015477", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.0, s=38.58, n=193, compute CI."}]}, "response": "95% CI ≈ [43.56, 54.44].", "meta": {"id": "WUA_STAT_25K_015478", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.21, s=9.78, n=17, compute CI."}]}, "response": "95% CI ≈ [18.56, 27.86].", "meta": {"id": "WUA_STAT_25K_015479", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.76, s=11.39, n=162, compute CI."}]}, "response": "95% CI ≈ [149.01, 152.51].", "meta": {"id": "WUA_STAT_25K_015480", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.39, compute E and Var."}]}, "response": "E[X]=4.29; Var[X]=2.617.", "meta": {"id": "WUA_STAT_25K_015481", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.85, μ=81.35, σ=14.73, compute z."}]}, "response": "z = -1.595.", "meta": {"id": "WUA_STAT_25K_015482", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.36, s=18.8, n=167, compute CI."}]}, "response": "95% CI ≈ [77.51, 83.21].", "meta": {"id": "WUA_STAT_25K_015483", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.28, s=7.18, n=32, compute CI."}]}, "response": "95% CI ≈ [51.79, 56.77].", "meta": {"id": "WUA_STAT_25K_015484", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.43, μ=44.36, σ=21.45, compute z."}]}, "response": "z = 1.169.", "meta": {"id": "WUA_STAT_25K_015485", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.162, sens=0.87, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.470.", "meta": {"id": "WUA_STAT_25K_015486", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=168.09, s=11.46, n=63, compute CI."}]}, "response": "95% CI ≈ [165.26, 170.92].", "meta": {"id": "WUA_STAT_25K_015487", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.66, compute E and Var."}]}, "response": "E[X]=20.46; Var[X]=6.956.", "meta": {"id": "WUA_STAT_25K_015488", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.104, sens=0.86, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.833.", "meta": {"id": "WUA_STAT_25K_015489", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.48, s=9.55, n=146, compute CI."}]}, "response": "95% CI ≈ [63.93, 67.03].", "meta": {"id": "WUA_STAT_25K_015490", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.19, s=12.61, n=40, compute CI."}]}, "response": "95% CI ≈ [103.28, 111.10].", "meta": {"id": "WUA_STAT_25K_015491", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.93, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.528.", "meta": {"id": "WUA_STAT_25K_015492", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.63, compute E and Var."}]}, "response": "E[X]=9.45; Var[X]=3.496.", "meta": {"id": "WUA_STAT_25K_015493", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.46, compute E and Var."}]}, "response": "E[X]=12.42; Var[X]=6.707.", "meta": {"id": "WUA_STAT_25K_015494", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.06, compute E and Var."}]}, "response": "E[X]=1.86; Var[X]=1.748.", "meta": {"id": "WUA_STAT_25K_015495", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.16, s=13.98, n=181, compute CI."}]}, "response": "95% CI ≈ [41.12, 45.20].", "meta": {"id": "WUA_STAT_25K_015496", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.35, s=34.75, n=147, compute CI."}]}, "response": "95% CI ≈ [47.73, 58.97].", "meta": {"id": "WUA_STAT_25K_015497", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.53, μ=70.1, σ=24.49, compute z."}]}, "response": "z = 1.079.", "meta": {"id": "WUA_STAT_25K_015498", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.2, compute E and Var."}]}, "response": "E[X]=5.00; Var[X]=4.000.", "meta": {"id": "WUA_STAT_25K_015499", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.14, compute E and Var."}]}, "response": "E[X]=4.48; Var[X]=3.853.", "meta": {"id": "WUA_STAT_25K_015500", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.28, compute E and Var."}]}, "response": "E[X]=3.92; Var[X]=2.822.", "meta": {"id": "WUA_STAT_25K_015501", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.43, compute E and Var."}]}, "response": "E[X]=15.91; Var[X]=9.069.", "meta": {"id": "WUA_STAT_25K_015502", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.73, compute E and Var."}]}, "response": "E[X]=29.20; Var[X]=7.884.", "meta": {"id": "WUA_STAT_25K_015503", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.49, s=37.12, n=52, compute CI."}]}, "response": "95% CI ≈ [43.40, 63.58].", "meta": {"id": "WUA_STAT_25K_015504", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.72, compute E and Var."}]}, "response": "E[X]=27.36; Var[X]=7.661.", "meta": {"id": "WUA_STAT_25K_015505", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.98, s=31.85, n=182, compute CI."}]}, "response": "95% CI ≈ [9.35, 18.61].", "meta": {"id": "WUA_STAT_25K_015506", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.066, sens=0.75, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.346.", "meta": {"id": "WUA_STAT_25K_015507", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.43, compute E and Var."}]}, "response": "E[X]=12.90; Var[X]=7.353.", "meta": {"id": "WUA_STAT_25K_015508", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=82.1, s=27.7, n=97, compute CI."}]}, "response": "95% CI ≈ [76.59, 87.61].", "meta": {"id": "WUA_STAT_25K_015509", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.9, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.516.", "meta": {"id": "WUA_STAT_25K_015510", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.08, compute E and Var."}]}, "response": "E[X]=3.20; Var[X]=2.944.", "meta": {"id": "WUA_STAT_25K_015511", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.58, compute E and Var."}]}, "response": "E[X]=18.56; Var[X]=7.795.", "meta": {"id": "WUA_STAT_25K_015512", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.16, μ=99.84, σ=11.15, compute z."}]}, "response": "z = -2.213.", "meta": {"id": "WUA_STAT_25K_015513", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.7, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.743.", "meta": {"id": "WUA_STAT_25K_015514", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.143, sens=0.75, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.439.", "meta": {"id": "WUA_STAT_25K_015515", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.5, μ=4.89, σ=9.78, compute z."}]}, "response": "z = 2.721.", "meta": {"id": "WUA_STAT_25K_015516", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.81, μ=85.71, σ=13.12, compute z."}]}, "response": "z = 0.008.", "meta": {"id": "WUA_STAT_25K_015517", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.52, compute E and Var."}]}, "response": "E[X]=3.64; Var[X]=1.747.", "meta": {"id": "WUA_STAT_25K_015518", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.97, s=27.53, n=198, compute CI."}]}, "response": "95% CI ≈ [64.14, 71.80].", "meta": {"id": "WUA_STAT_25K_015519", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.096, sens=0.87, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.262.", "meta": {"id": "WUA_STAT_25K_015520", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.07, sens=0.71, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.572.", "meta": {"id": "WUA_STAT_25K_015521", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.217, sens=0.99, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.555.", "meta": {"id": "WUA_STAT_25K_015522", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.051, sens=0.95, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.282.", "meta": {"id": "WUA_STAT_25K_015523", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.5, compute E and Var."}]}, "response": "E[X]=20.00; Var[X]=10.000.", "meta": {"id": "WUA_STAT_25K_015524", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.33, s=27.51, n=43, compute CI."}]}, "response": "95% CI ≈ [36.11, 52.55].", "meta": {"id": "WUA_STAT_25K_015525", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.16, compute E and Var."}]}, "response": "E[X]=4.32; Var[X]=3.629.", "meta": {"id": "WUA_STAT_25K_015526", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.124, sens=0.84, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.748.", "meta": {"id": "WUA_STAT_25K_015527", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.17, compute E and Var."}]}, "response": "E[X]=5.61; Var[X]=4.656.", "meta": {"id": "WUA_STAT_25K_015528", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.41, compute E and Var."}]}, "response": "E[X]=10.66; Var[X]=6.289.", "meta": {"id": "WUA_STAT_25K_015529", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.76, compute E and Var."}]}, "response": "E[X]=25.08; Var[X]=6.019.", "meta": {"id": "WUA_STAT_25K_015530", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=2.83, μ=28.82, σ=8.52, compute z."}]}, "response": "z = -3.050.", "meta": {"id": "WUA_STAT_25K_015531", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=63.12, s=9.41, n=45, compute CI."}]}, "response": "95% CI ≈ [60.37, 65.87].", "meta": {"id": "WUA_STAT_25K_015532", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.64, μ=24.9, σ=22.29, compute z."}]}, "response": "z = -0.415.", "meta": {"id": "WUA_STAT_25K_015533", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.98, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.228.", "meta": {"id": "WUA_STAT_25K_015534", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.47, s=32.12, n=169, compute CI."}]}, "response": "95% CI ≈ [38.63, 48.31].", "meta": {"id": "WUA_STAT_25K_015535", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.94, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.470.", "meta": {"id": "WUA_STAT_25K_015536", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.6, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=2.880.", "meta": {"id": "WUA_STAT_25K_015537", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.93, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.878.", "meta": {"id": "WUA_STAT_25K_015538", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.01, μ=19.19, σ=20.68, compute z."}]}, "response": "z = 1.732.", "meta": {"id": "WUA_STAT_25K_015539", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.99, s=5.17, n=19, compute CI."}]}, "response": "95% CI ≈ [161.67, 166.31].", "meta": {"id": "WUA_STAT_25K_015540", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.48, compute E and Var."}]}, "response": "E[X]=12.96; Var[X]=6.739.", "meta": {"id": "WUA_STAT_25K_015541", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.53, s=15.97, n=115, compute CI."}]}, "response": "95% CI ≈ [73.61, 79.45].", "meta": {"id": "WUA_STAT_25K_015542", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.22, s=18.68, n=167, compute CI."}]}, "response": "95% CI ≈ [31.39, 37.05].", "meta": {"id": "WUA_STAT_25K_015543", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.83, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.609.", "meta": {"id": "WUA_STAT_25K_015544", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.07, μ=95.07, σ=2.35, compute z."}]}, "response": "z = -11.915.", "meta": {"id": "WUA_STAT_25K_015545", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.48, compute E and Var."}]}, "response": "E[X]=4.80; Var[X]=2.496.", "meta": {"id": "WUA_STAT_25K_015546", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.54, s=36.97, n=136, compute CI."}]}, "response": "95% CI ≈ [51.33, 63.75].", "meta": {"id": "WUA_STAT_25K_015547", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.06, μ=66.52, σ=5.52, compute z."}]}, "response": "z = 3.540.", "meta": {"id": "WUA_STAT_25K_015548", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.1, μ=3.06, σ=9.75, compute z."}]}, "response": "z = 0.414.", "meta": {"id": "WUA_STAT_25K_015549", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.8, s=21.39, n=45, compute CI."}]}, "response": "95% CI ≈ [169.55, 182.05].", "meta": {"id": "WUA_STAT_25K_015550", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.26, compute E and Var."}]}, "response": "E[X]=2.08; Var[X]=1.539.", "meta": {"id": "WUA_STAT_25K_015551", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=96.29, s=33.43, n=49, compute CI."}]}, "response": "95% CI ≈ [86.93, 105.65].", "meta": {"id": "WUA_STAT_25K_015552", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=51.49, s=3.78, n=49, compute CI."}]}, "response": "95% CI ≈ [50.43, 52.55].", "meta": {"id": "WUA_STAT_25K_015553", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.43, μ=15.05, σ=24.67, compute z."}]}, "response": "z = 1.637.", "meta": {"id": "WUA_STAT_25K_015554", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=69.44, μ=14.75, σ=24.59, compute z."}]}, "response": "z = 2.224.", "meta": {"id": "WUA_STAT_25K_015555", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.27, s=39.89, n=193, compute CI."}]}, "response": "95% CI ≈ [150.64, 161.90].", "meta": {"id": "WUA_STAT_25K_015556", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.62, s=14.01, n=55, compute CI."}]}, "response": "95% CI ≈ [186.92, 194.32].", "meta": {"id": "WUA_STAT_25K_015557", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.107, sens=0.87, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.271.", "meta": {"id": "WUA_STAT_25K_015558", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.0, s=20.82, n=105, compute CI."}]}, "response": "95% CI ≈ [132.02, 139.98].", "meta": {"id": "WUA_STAT_25K_015559", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.42, μ=1.12, σ=17.99, compute z."}]}, "response": "z = 5.186.", "meta": {"id": "WUA_STAT_25K_015560", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.09, compute E and Var."}]}, "response": "E[X]=2.70; Var[X]=2.457.", "meta": {"id": "WUA_STAT_25K_015561", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.52, compute E and Var."}]}, "response": "E[X]=5.20; Var[X]=2.496.", "meta": {"id": "WUA_STAT_25K_015562", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.094, sens=0.98, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.772.", "meta": {"id": "WUA_STAT_25K_015563", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.33, compute E and Var."}]}, "response": "E[X]=6.60; Var[X]=4.422.", "meta": {"id": "WUA_STAT_25K_015564", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=161.82, s=21.56, n=169, compute CI."}]}, "response": "95% CI ≈ [158.57, 165.07].", "meta": {"id": "WUA_STAT_25K_015565", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.87, μ=98.99, σ=13.36, compute z."}]}, "response": "z = -6.671.", "meta": {"id": "WUA_STAT_25K_015566", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.49, compute E and Var."}]}, "response": "E[X]=11.27; Var[X]=5.748.", "meta": {"id": "WUA_STAT_25K_015567", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.51, s=19.12, n=200, compute CI."}]}, "response": "95% CI ≈ [118.86, 124.16].", "meta": {"id": "WUA_STAT_25K_015568", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.01, μ=88.38, σ=20.47, compute z."}]}, "response": "z = 0.519.", "meta": {"id": "WUA_STAT_25K_015569", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.8, s=35.25, n=93, compute CI."}]}, "response": "95% CI ≈ [124.64, 138.96].", "meta": {"id": "WUA_STAT_25K_015570", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.168, sens=0.87, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.539.", "meta": {"id": "WUA_STAT_25K_015571", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.78, μ=9.67, σ=24.06, compute z."}]}, "response": "z = 1.418.", "meta": {"id": "WUA_STAT_25K_015572", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.227, sens=0.77, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.883.", "meta": {"id": "WUA_STAT_25K_015573", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.41, compute E and Var."}]}, "response": "E[X]=15.99; Var[X]=9.434.", "meta": {"id": "WUA_STAT_25K_015574", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.58, compute E and Var."}]}, "response": "E[X]=14.50; Var[X]=6.090.", "meta": {"id": "WUA_STAT_25K_015575", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.092, sens=0.77, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.661.", "meta": {"id": "WUA_STAT_25K_015576", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.95, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.856.", "meta": {"id": "WUA_STAT_25K_015577", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.31, μ=66.6, σ=14.77, compute z."}]}, "response": "z = -2.728.", "meta": {"id": "WUA_STAT_25K_015578", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.2, compute E and Var."}]}, "response": "E[X]=2.60; Var[X]=2.080.", "meta": {"id": "WUA_STAT_25K_015579", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=197.5, s=20.05, n=60, compute CI."}]}, "response": "95% CI ≈ [192.43, 202.57].", "meta": {"id": "WUA_STAT_25K_015580", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.87, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.302.", "meta": {"id": "WUA_STAT_25K_015581", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.129, sens=0.75, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.308.", "meta": {"id": "WUA_STAT_25K_015582", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.66, s=16.86, n=115, compute CI."}]}, "response": "95% CI ≈ [84.58, 90.74].", "meta": {"id": "WUA_STAT_25K_015583", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.33, compute E and Var."}]}, "response": "E[X]=2.64; Var[X]=1.769.", "meta": {"id": "WUA_STAT_25K_015584", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.3, s=3.19, n=168, compute CI."}]}, "response": "95% CI ≈ [39.82, 40.78].", "meta": {"id": "WUA_STAT_25K_015585", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.073, sens=0.88, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.224.", "meta": {"id": "WUA_STAT_25K_015586", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.78, μ=72.34, σ=15.3, compute z."}]}, "response": "z = 0.682.", "meta": {"id": "WUA_STAT_25K_015587", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.24, μ=76.69, σ=23.02, compute z."}]}, "response": "z = -0.541.", "meta": {"id": "WUA_STAT_25K_015588", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=104.22, s=35.38, n=91, compute CI."}]}, "response": "95% CI ≈ [96.95, 111.49].", "meta": {"id": "WUA_STAT_25K_015589", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.13, μ=35.88, σ=15.04, compute z."}]}, "response": "z = -1.446.", "meta": {"id": "WUA_STAT_25K_015590", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.85, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.462.", "meta": {"id": "WUA_STAT_25K_015591", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.72, μ=12.94, σ=14.24, compute z."}]}, "response": "z = 1.951.", "meta": {"id": "WUA_STAT_25K_015592", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.204, sens=0.79, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.609.", "meta": {"id": "WUA_STAT_25K_015593", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.79, s=7.59, n=127, compute CI."}]}, "response": "95% CI ≈ [109.47, 112.11].", "meta": {"id": "WUA_STAT_25K_015594", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.76, μ=94.01, σ=1.3, compute z."}]}, "response": "z = -2.500.", "meta": {"id": "WUA_STAT_25K_015595", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.77, s=21.78, n=152, compute CI."}]}, "response": "95% CI ≈ [87.31, 94.23].", "meta": {"id": "WUA_STAT_25K_015596", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=83.13, μ=22.99, σ=15.41, compute z."}]}, "response": "z = 3.903.", "meta": {"id": "WUA_STAT_25K_015597", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.8, s=31.63, n=81, compute CI."}]}, "response": "95% CI ≈ [23.91, 37.69].", "meta": {"id": "WUA_STAT_25K_015598", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.84, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.402.", "meta": {"id": "WUA_STAT_25K_015599", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.79, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.335.", "meta": {"id": "WUA_STAT_25K_015600", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.03, s=9.06, n=42, compute CI."}]}, "response": "95% CI ≈ [58.29, 63.77].", "meta": {"id": "WUA_STAT_25K_015601", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.62, s=7.89, n=55, compute CI."}]}, "response": "95% CI ≈ [119.53, 123.71].", "meta": {"id": "WUA_STAT_25K_015602", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.38, s=16.76, n=82, compute CI."}]}, "response": "95% CI ≈ [23.75, 31.01].", "meta": {"id": "WUA_STAT_25K_015603", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.43, s=25.13, n=50, compute CI."}]}, "response": "95% CI ≈ [173.46, 187.40].", "meta": {"id": "WUA_STAT_25K_015604", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.36, compute E and Var."}]}, "response": "E[X]=11.16; Var[X]=7.142.", "meta": {"id": "WUA_STAT_25K_015605", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.48, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=3.744.", "meta": {"id": "WUA_STAT_25K_015606", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.38, s=23.53, n=66, compute CI."}]}, "response": "95% CI ≈ [107.70, 119.06].", "meta": {"id": "WUA_STAT_25K_015607", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.92, μ=69.74, σ=8.09, compute z."}]}, "response": "z = -0.472.", "meta": {"id": "WUA_STAT_25K_015608", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.11, s=7.97, n=174, compute CI."}]}, "response": "95% CI ≈ [33.93, 36.29].", "meta": {"id": "WUA_STAT_25K_015609", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.36, μ=41.54, σ=4.64, compute z."}]}, "response": "z = 9.444.", "meta": {"id": "WUA_STAT_25K_015610", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.93, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.463.", "meta": {"id": "WUA_STAT_25K_015611", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.178, sens=0.91, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.422.", "meta": {"id": "WUA_STAT_25K_015612", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.51, compute E and Var."}]}, "response": "E[X]=14.79; Var[X]=7.247.", "meta": {"id": "WUA_STAT_25K_015613", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.46, μ=87.9, σ=19.3, compute z."}]}, "response": "z = -1.111.", "meta": {"id": "WUA_STAT_25K_015614", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.06, compute E and Var."}]}, "response": "E[X]=0.72; Var[X]=0.677.", "meta": {"id": "WUA_STAT_25K_015615", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.171, sens=0.74, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.792.", "meta": {"id": "WUA_STAT_25K_015616", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.14, compute E and Var."}]}, "response": "E[X]=1.54; Var[X]=1.324.", "meta": {"id": "WUA_STAT_25K_015617", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.18, sens=0.72, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.429.", "meta": {"id": "WUA_STAT_25K_015618", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.22, compute E and Var."}]}, "response": "E[X]=1.10; Var[X]=0.858.", "meta": {"id": "WUA_STAT_25K_015619", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.55, μ=37.1, σ=21.06, compute z."}]}, "response": "z = 1.303.", "meta": {"id": "WUA_STAT_25K_015620", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.39, μ=62.01, σ=23.44, compute z."}]}, "response": "z = 1.040.", "meta": {"id": "WUA_STAT_25K_015621", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=14.92, s=37.66, n=81, compute CI."}]}, "response": "95% CI ≈ [6.72, 23.12].", "meta": {"id": "WUA_STAT_25K_015622", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.36, compute E and Var."}]}, "response": "E[X]=8.64; Var[X]=5.530.", "meta": {"id": "WUA_STAT_25K_015623", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.05, s=14.37, n=80, compute CI."}]}, "response": "95% CI ≈ [55.90, 62.20].", "meta": {"id": "WUA_STAT_25K_015624", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.15, s=25.25, n=75, compute CI."}]}, "response": "95% CI ≈ [145.44, 156.86].", "meta": {"id": "WUA_STAT_25K_015625", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.4, s=27.76, n=138, compute CI."}]}, "response": "95% CI ≈ [20.77, 30.03].", "meta": {"id": "WUA_STAT_25K_015626", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.17, compute E and Var."}]}, "response": "E[X]=6.80; Var[X]=5.644.", "meta": {"id": "WUA_STAT_25K_015627", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.16, μ=43.17, σ=12.9, compute z."}]}, "response": "z = 1.627.", "meta": {"id": "WUA_STAT_25K_015628", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.021, sens=0.92, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.104.", "meta": {"id": "WUA_STAT_25K_015629", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.05, compute E and Var."}]}, "response": "E[X]=1.20; Var[X]=1.140.", "meta": {"id": "WUA_STAT_25K_015630", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=56.64, s=28.49, n=140, compute CI."}]}, "response": "95% CI ≈ [51.92, 61.36].", "meta": {"id": "WUA_STAT_25K_015631", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.86, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.579.", "meta": {"id": "WUA_STAT_25K_015632", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.07, s=32.21, n=20, compute CI."}]}, "response": "95% CI ≈ [62.95, 91.19].", "meta": {"id": "WUA_STAT_25K_015633", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.42, compute E and Var."}]}, "response": "E[X]=10.08; Var[X]=5.846.", "meta": {"id": "WUA_STAT_25K_015634", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.88, μ=65.35, σ=4.49, compute z."}]}, "response": "z = -3.668.", "meta": {"id": "WUA_STAT_25K_015635", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.071, sens=0.84, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.682.", "meta": {"id": "WUA_STAT_25K_015636", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=95.63, s=7.07, n=192, compute CI."}]}, "response": "95% CI ≈ [94.63, 96.63].", "meta": {"id": "WUA_STAT_25K_015637", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.92, μ=11.12, σ=13.15, compute z."}]}, "response": "z = 4.624.", "meta": {"id": "WUA_STAT_25K_015638", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.89, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.134.", "meta": {"id": "WUA_STAT_25K_015639", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.37, s=2.7, n=37, compute CI."}]}, "response": "95% CI ≈ [21.50, 23.24].", "meta": {"id": "WUA_STAT_25K_015640", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.16, s=37.05, n=190, compute CI."}]}, "response": "95% CI ≈ [123.89, 134.43].", "meta": {"id": "WUA_STAT_25K_015641", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.8, s=22.86, n=64, compute CI."}]}, "response": "95% CI ≈ [69.20, 80.40].", "meta": {"id": "WUA_STAT_25K_015642", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.228, sens=0.79, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.642.", "meta": {"id": "WUA_STAT_25K_015643", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.46, compute E and Var."}]}, "response": "E[X]=8.74; Var[X]=4.720.", "meta": {"id": "WUA_STAT_25K_015644", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=91.31, s=36.62, n=179, compute CI."}]}, "response": "95% CI ≈ [85.95, 96.67].", "meta": {"id": "WUA_STAT_25K_015645", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=94.43, s=8.15, n=90, compute CI."}]}, "response": "95% CI ≈ [92.75, 96.11].", "meta": {"id": "WUA_STAT_25K_015646", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.27, s=24.82, n=98, compute CI."}]}, "response": "95% CI ≈ [106.36, 116.18].", "meta": {"id": "WUA_STAT_25K_015647", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.58, compute E and Var."}]}, "response": "E[X]=11.02; Var[X]=4.628.", "meta": {"id": "WUA_STAT_25K_015648", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=132.64, s=20.14, n=200, compute CI."}]}, "response": "95% CI ≈ [129.85, 135.43].", "meta": {"id": "WUA_STAT_25K_015649", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.193, sens=0.76, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.722.", "meta": {"id": "WUA_STAT_25K_015650", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.29, s=26.09, n=67, compute CI."}]}, "response": "95% CI ≈ [133.04, 145.54].", "meta": {"id": "WUA_STAT_25K_015651", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.032, sens=0.72, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.106.", "meta": {"id": "WUA_STAT_25K_015652", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.77, compute E and Var."}]}, "response": "E[X]=16.94; Var[X]=3.896.", "meta": {"id": "WUA_STAT_25K_015653", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.4, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=4.320.", "meta": {"id": "WUA_STAT_25K_015654", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.63, s=5.43, n=52, compute CI."}]}, "response": "95% CI ≈ [35.15, 38.11].", "meta": {"id": "WUA_STAT_25K_015655", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.57, compute E and Var."}]}, "response": "E[X]=5.13; Var[X]=2.206.", "meta": {"id": "WUA_STAT_25K_015656", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.97, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.460.", "meta": {"id": "WUA_STAT_25K_015657", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.54, compute E and Var."}]}, "response": "E[X]=13.50; Var[X]=6.210.", "meta": {"id": "WUA_STAT_25K_015658", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.11, compute E and Var."}]}, "response": "E[X]=3.30; Var[X]=2.937.", "meta": {"id": "WUA_STAT_25K_015659", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.31, μ=93.27, σ=12.79, compute z."}]}, "response": "z = -2.108.", "meta": {"id": "WUA_STAT_25K_015660", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=96.97, s=38.37, n=141, compute CI."}]}, "response": "95% CI ≈ [90.64, 103.30].", "meta": {"id": "WUA_STAT_25K_015661", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.8, compute E and Var."}]}, "response": "E[X]=4.00; Var[X]=0.800.", "meta": {"id": "WUA_STAT_25K_015662", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.04, μ=1.85, σ=18.38, compute z."}]}, "response": "z = 2.187.", "meta": {"id": "WUA_STAT_25K_015663", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.71, μ=28.88, σ=8.83, compute z."}]}, "response": "z = 3.492.", "meta": {"id": "WUA_STAT_25K_015664", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=105.63, s=35.51, n=174, compute CI."}]}, "response": "95% CI ≈ [100.35, 110.91].", "meta": {"id": "WUA_STAT_25K_015665", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.63, μ=90.83, σ=16.2, compute z."}]}, "response": "z = -0.691.", "meta": {"id": "WUA_STAT_25K_015666", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.85, μ=73.18, σ=2.04, compute z."}]}, "response": "z = 1.799.", "meta": {"id": "WUA_STAT_25K_015667", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.82, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.603.", "meta": {"id": "WUA_STAT_25K_015668", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.055, sens=0.89, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.161.", "meta": {"id": "WUA_STAT_25K_015669", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.93, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.859.", "meta": {"id": "WUA_STAT_25K_015670", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.92, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.374.", "meta": {"id": "WUA_STAT_25K_015671", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.79, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.253.", "meta": {"id": "WUA_STAT_25K_015672", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.24, compute E and Var."}]}, "response": "E[X]=3.60; Var[X]=2.736.", "meta": {"id": "WUA_STAT_25K_015673", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.24, μ=93.48, σ=3.9, compute z."}]}, "response": "z = -20.318.", "meta": {"id": "WUA_STAT_25K_015674", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=100.31, s=1.3, n=14, compute CI."}]}, "response": "95% CI ≈ [99.63, 100.99].", "meta": {"id": "WUA_STAT_25K_015675", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.15, compute E and Var."}]}, "response": "E[X]=0.90; Var[X]=0.765.", "meta": {"id": "WUA_STAT_25K_015676", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.88, μ=58.49, σ=6.56, compute z."}]}, "response": "z = -1.160.", "meta": {"id": "WUA_STAT_25K_015677", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.7, compute E and Var."}]}, "response": "E[X]=6.30; Var[X]=1.890.", "meta": {"id": "WUA_STAT_25K_015678", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.74, s=26.09, n=137, compute CI."}]}, "response": "95% CI ≈ [18.37, 27.11].", "meta": {"id": "WUA_STAT_25K_015679", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.83, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.577.", "meta": {"id": "WUA_STAT_25K_015680", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.9, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.805.", "meta": {"id": "WUA_STAT_25K_015681", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.58, compute E and Var."}]}, "response": "E[X]=8.12; Var[X]=3.410.", "meta": {"id": "WUA_STAT_25K_015682", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.233, sens=0.86, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.474.", "meta": {"id": "WUA_STAT_25K_015683", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.66, compute E and Var."}]}, "response": "E[X]=9.90; Var[X]=3.366.", "meta": {"id": "WUA_STAT_25K_015684", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.13, s=17.82, n=14, compute CI."}]}, "response": "95% CI ≈ [180.80, 199.46].", "meta": {"id": "WUA_STAT_25K_015685", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.1, s=5.5, n=197, compute CI."}]}, "response": "95% CI ≈ [42.33, 43.87].", "meta": {"id": "WUA_STAT_25K_015686", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.79, μ=54.48, σ=11.43, compute z."}]}, "response": "z = -1.898.", "meta": {"id": "WUA_STAT_25K_015687", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.71, s=33.17, n=29, compute CI."}]}, "response": "95% CI ≈ [1.64, 25.78].", "meta": {"id": "WUA_STAT_25K_015688", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.58, compute E and Var."}]}, "response": "E[X]=11.60; Var[X]=4.872.", "meta": {"id": "WUA_STAT_25K_015689", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=16.3, s=13.02, n=10, compute CI."}]}, "response": "95% CI ≈ [8.23, 24.37].", "meta": {"id": "WUA_STAT_25K_015690", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.78, compute E and Var."}]}, "response": "E[X]=27.30; Var[X]=6.006.", "meta": {"id": "WUA_STAT_25K_015691", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.165, sens=0.99, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.662.", "meta": {"id": "WUA_STAT_25K_015692", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.158, sens=0.88, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.478.", "meta": {"id": "WUA_STAT_25K_015693", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.74, μ=95.72, σ=16.71, compute z."}]}, "response": "z = -5.145.", "meta": {"id": "WUA_STAT_25K_015694", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.89, s=25.16, n=52, compute CI."}]}, "response": "95% CI ≈ [37.05, 50.73].", "meta": {"id": "WUA_STAT_25K_015695", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.71, compute E and Var."}]}, "response": "E[X]=8.52; Var[X]=2.471.", "meta": {"id": "WUA_STAT_25K_015696", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.51, μ=12.94, σ=14.48, compute z."}]}, "response": "z = 3.216.", "meta": {"id": "WUA_STAT_25K_015697", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.14, compute E and Var."}]}, "response": "E[X]=0.98; Var[X]=0.843.", "meta": {"id": "WUA_STAT_25K_015698", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.165, sens=0.91, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.857.", "meta": {"id": "WUA_STAT_25K_015699", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.67, μ=70.17, σ=5.21, compute z."}]}, "response": "z = 2.975.", "meta": {"id": "WUA_STAT_25K_015700", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.02, μ=36.77, σ=22.02, compute z."}]}, "response": "z = -0.261.", "meta": {"id": "WUA_STAT_25K_015701", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.55, compute E and Var."}]}, "response": "E[X]=17.05; Var[X]=7.672.", "meta": {"id": "WUA_STAT_25K_015702", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.119, sens=0.94, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.927.", "meta": {"id": "WUA_STAT_25K_015703", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.232, sens=0.74, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.882.", "meta": {"id": "WUA_STAT_25K_015704", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.17, μ=33.05, σ=13.9, compute z."}]}, "response": "z = 1.088.", "meta": {"id": "WUA_STAT_25K_015705", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.226, sens=0.76, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.847.", "meta": {"id": "WUA_STAT_25K_015706", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.6, s=3.3, n=139, compute CI."}]}, "response": "95% CI ≈ [127.05, 128.15].", "meta": {"id": "WUA_STAT_25K_015707", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.74, compute E and Var."}]}, "response": "E[X]=23.68; Var[X]=6.157.", "meta": {"id": "WUA_STAT_25K_015708", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=22.07, μ=68.58, σ=15.01, compute z."}]}, "response": "z = -3.099.", "meta": {"id": "WUA_STAT_25K_015709", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.46, s=31.82, n=137, compute CI."}]}, "response": "95% CI ≈ [161.13, 171.79].", "meta": {"id": "WUA_STAT_25K_015710", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.81, s=21.05, n=43, compute CI."}]}, "response": "95% CI ≈ [27.52, 40.10].", "meta": {"id": "WUA_STAT_25K_015711", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=91.96, s=21.48, n=37, compute CI."}]}, "response": "95% CI ≈ [85.04, 98.88].", "meta": {"id": "WUA_STAT_25K_015712", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.73, μ=47.05, σ=15.33, compute z."}]}, "response": "z = -1.260.", "meta": {"id": "WUA_STAT_25K_015713", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.07, sens=0.77, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.167.", "meta": {"id": "WUA_STAT_25K_015714", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=196.79, s=17.92, n=17, compute CI."}]}, "response": "95% CI ≈ [188.27, 205.31].", "meta": {"id": "WUA_STAT_25K_015715", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.15, μ=16.64, σ=24.74, compute z."}]}, "response": "z = -0.505.", "meta": {"id": "WUA_STAT_25K_015716", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.41, s=6.14, n=56, compute CI."}]}, "response": "95% CI ≈ [110.80, 114.02].", "meta": {"id": "WUA_STAT_25K_015717", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.55, μ=20.93, σ=23.07, compute z."}]}, "response": "z = 0.720.", "meta": {"id": "WUA_STAT_25K_015718", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.9, μ=35.04, σ=19.15, compute z."}]}, "response": "z = -0.425.", "meta": {"id": "WUA_STAT_25K_015719", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.27, compute E and Var."}]}, "response": "E[X]=6.21; Var[X]=4.533.", "meta": {"id": "WUA_STAT_25K_015720", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.125, sens=0.79, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.414.", "meta": {"id": "WUA_STAT_25K_015721", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=39.75, s=39.72, n=110, compute CI."}]}, "response": "95% CI ≈ [32.33, 47.17].", "meta": {"id": "WUA_STAT_25K_015722", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.059, sens=0.75, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.320.", "meta": {"id": "WUA_STAT_25K_015723", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.12, compute E and Var."}]}, "response": "E[X]=3.00; Var[X]=2.640.", "meta": {"id": "WUA_STAT_25K_015724", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.81, s=29.66, n=193, compute CI."}]}, "response": "95% CI ≈ [66.63, 74.99].", "meta": {"id": "WUA_STAT_25K_015725", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.071, sens=0.84, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.211.", "meta": {"id": "WUA_STAT_25K_015726", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.29, compute E and Var."}]}, "response": "E[X]=8.12; Var[X]=5.765.", "meta": {"id": "WUA_STAT_25K_015727", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.219, sens=0.89, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.757.", "meta": {"id": "WUA_STAT_25K_015728", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.93, μ=9.79, σ=16.03, compute z."}]}, "response": "z = 0.445.", "meta": {"id": "WUA_STAT_25K_015729", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=100.54, s=22.15, n=145, compute CI."}]}, "response": "95% CI ≈ [96.93, 104.15].", "meta": {"id": "WUA_STAT_25K_015730", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.28, s=12.02, n=118, compute CI."}]}, "response": "95% CI ≈ [34.11, 38.45].", "meta": {"id": "WUA_STAT_25K_015731", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.29, compute E and Var."}]}, "response": "E[X]=10.73; Var[X]=7.618.", "meta": {"id": "WUA_STAT_25K_015732", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.65, compute E and Var."}]}, "response": "E[X]=3.25; Var[X]=1.137.", "meta": {"id": "WUA_STAT_25K_015733", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.064, sens=0.92, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.386.", "meta": {"id": "WUA_STAT_25K_015734", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.81, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.658.", "meta": {"id": "WUA_STAT_25K_015735", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.03, sens=0.94, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.744.", "meta": {"id": "WUA_STAT_25K_015736", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.95, s=2.01, n=114, compute CI."}]}, "response": "95% CI ≈ [34.58, 35.32].", "meta": {"id": "WUA_STAT_25K_015737", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.241, sens=0.72, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.851.", "meta": {"id": "WUA_STAT_25K_015738", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.246, sens=0.71, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.853.", "meta": {"id": "WUA_STAT_25K_015739", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.31, μ=36.19, σ=16.69, compute z."}]}, "response": "z = -1.611.", "meta": {"id": "WUA_STAT_25K_015740", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.19, μ=22.24, σ=22.18, compute z."}]}, "response": "z = 1.666.", "meta": {"id": "WUA_STAT_25K_015741", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.08, compute E and Var."}]}, "response": "E[X]=1.36; Var[X]=1.251.", "meta": {"id": "WUA_STAT_25K_015742", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.46, s=37.81, n=43, compute CI."}]}, "response": "95% CI ≈ [13.16, 35.76].", "meta": {"id": "WUA_STAT_25K_015743", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=21.3, s=1.39, n=93, compute CI."}]}, "response": "95% CI ≈ [21.02, 21.58].", "meta": {"id": "WUA_STAT_25K_015744", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.137, sens=0.94, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.713.", "meta": {"id": "WUA_STAT_25K_015745", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.58, compute E and Var."}]}, "response": "E[X]=2.90; Var[X]=1.218.", "meta": {"id": "WUA_STAT_25K_015746", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.87, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.500.", "meta": {"id": "WUA_STAT_25K_015747", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.6, s=33.25, n=164, compute CI."}]}, "response": "95% CI ≈ [194.51, 204.69].", "meta": {"id": "WUA_STAT_25K_015748", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.7, μ=24.78, σ=21.27, compute z."}]}, "response": "z = 1.407.", "meta": {"id": "WUA_STAT_25K_015749", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.34, s=6.42, n=53, compute CI."}]}, "response": "95% CI ≈ [180.61, 184.07].", "meta": {"id": "WUA_STAT_25K_015750", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.37, s=35.9, n=96, compute CI."}]}, "response": "95% CI ≈ [31.19, 45.55].", "meta": {"id": "WUA_STAT_25K_015751", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.3, compute E and Var."}]}, "response": "E[X]=8.70; Var[X]=6.090.", "meta": {"id": "WUA_STAT_25K_015752", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.19, μ=9.0, σ=1.7, compute z."}]}, "response": "z = 18.935.", "meta": {"id": "WUA_STAT_25K_015753", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.07, μ=10.42, σ=11.72, compute z."}]}, "response": "z = 2.786.", "meta": {"id": "WUA_STAT_25K_015754", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.16, μ=60.0, σ=4.06, compute z."}]}, "response": "z = 7.182.", "meta": {"id": "WUA_STAT_25K_015755", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.79, compute E and Var."}]}, "response": "E[X]=14.22; Var[X]=2.986.", "meta": {"id": "WUA_STAT_25K_015756", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=141.44, s=11.5, n=174, compute CI."}]}, "response": "95% CI ≈ [139.73, 143.15].", "meta": {"id": "WUA_STAT_25K_015757", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.67, compute E and Var."}]}, "response": "E[X]=6.70; Var[X]=2.211.", "meta": {"id": "WUA_STAT_25K_015758", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.89, s=36.5, n=25, compute CI."}]}, "response": "95% CI ≈ [161.58, 190.20].", "meta": {"id": "WUA_STAT_25K_015759", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.68, s=37.51, n=189, compute CI."}]}, "response": "95% CI ≈ [30.33, 41.03].", "meta": {"id": "WUA_STAT_25K_015760", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.226, sens=0.79, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.452.", "meta": {"id": "WUA_STAT_25K_015761", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.91, s=3.71, n=127, compute CI."}]}, "response": "95% CI ≈ [167.26, 168.56].", "meta": {"id": "WUA_STAT_25K_015762", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=162.09, s=10.08, n=12, compute CI."}]}, "response": "95% CI ≈ [156.39, 167.79].", "meta": {"id": "WUA_STAT_25K_015763", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.065, sens=0.97, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.200.", "meta": {"id": "WUA_STAT_25K_015764", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.09, s=14.5, n=35, compute CI."}]}, "response": "95% CI ≈ [73.29, 82.89].", "meta": {"id": "WUA_STAT_25K_015765", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.6, s=1.83, n=168, compute CI."}]}, "response": "95% CI ≈ [160.32, 160.88].", "meta": {"id": "WUA_STAT_25K_015766", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.27, μ=30.44, σ=11.71, compute z."}]}, "response": "z = 1.523.", "meta": {"id": "WUA_STAT_25K_015767", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=100.37, s=17.15, n=75, compute CI."}]}, "response": "95% CI ≈ [96.49, 104.25].", "meta": {"id": "WUA_STAT_25K_015768", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.71, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.429.", "meta": {"id": "WUA_STAT_25K_015769", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=160.5, s=17.81, n=27, compute CI."}]}, "response": "95% CI ≈ [153.78, 167.22].", "meta": {"id": "WUA_STAT_25K_015770", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=22.15, s=22.7, n=96, compute CI."}]}, "response": "95% CI ≈ [17.61, 26.69].", "meta": {"id": "WUA_STAT_25K_015771", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.25, compute E and Var."}]}, "response": "E[X]=7.50; Var[X]=5.625.", "meta": {"id": "WUA_STAT_25K_015772", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=27.62, s=21.5, n=114, compute CI."}]}, "response": "95% CI ≈ [23.67, 31.57].", "meta": {"id": "WUA_STAT_25K_015773", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.36, s=25.13, n=11, compute CI."}]}, "response": "95% CI ≈ [124.51, 154.21].", "meta": {"id": "WUA_STAT_25K_015774", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.41, compute E and Var."}]}, "response": "E[X]=12.30; Var[X]=7.257.", "meta": {"id": "WUA_STAT_25K_015775", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.19, μ=91.74, σ=18.85, compute z."}]}, "response": "z = -3.796.", "meta": {"id": "WUA_STAT_25K_015776", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.66, μ=3.34, σ=16.51, compute z."}]}, "response": "z = 5.531.", "meta": {"id": "WUA_STAT_25K_015777", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.6, μ=50.2, σ=2.46, compute z."}]}, "response": "z = -10.000.", "meta": {"id": "WUA_STAT_25K_015778", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.29, s=3.58, n=133, compute CI."}]}, "response": "95% CI ≈ [187.68, 188.90].", "meta": {"id": "WUA_STAT_25K_015779", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=108.0, s=13.44, n=182, compute CI."}]}, "response": "95% CI ≈ [106.05, 109.95].", "meta": {"id": "WUA_STAT_25K_015780", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.75, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.415.", "meta": {"id": "WUA_STAT_25K_015781", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.22, s=16.04, n=184, compute CI."}]}, "response": "95% CI ≈ [12.90, 17.54].", "meta": {"id": "WUA_STAT_25K_015782", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.87, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.662.", "meta": {"id": "WUA_STAT_25K_015783", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.73, μ=25.47, σ=7.52, compute z."}]}, "response": "z = 1.231.", "meta": {"id": "WUA_STAT_25K_015784", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.45, μ=82.59, σ=19.16, compute z."}]}, "response": "z = -2.095.", "meta": {"id": "WUA_STAT_25K_015785", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.78, compute E and Var."}]}, "response": "E[X]=28.86; Var[X]=6.349.", "meta": {"id": "WUA_STAT_25K_015786", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.78, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.610.", "meta": {"id": "WUA_STAT_25K_015787", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.65, μ=91.7, σ=20.55, compute z."}]}, "response": "z = -1.024.", "meta": {"id": "WUA_STAT_25K_015788", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.38, s=21.23, n=57, compute CI."}]}, "response": "95% CI ≈ [148.87, 159.89].", "meta": {"id": "WUA_STAT_25K_015789", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.19, compute E and Var."}]}, "response": "E[X]=7.22; Var[X]=5.848.", "meta": {"id": "WUA_STAT_25K_015790", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.62, compute E and Var."}]}, "response": "E[X]=11.78; Var[X]=4.476.", "meta": {"id": "WUA_STAT_25K_015791", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.1, μ=52.69, σ=4.36, compute z."}]}, "response": "z = -11.833.", "meta": {"id": "WUA_STAT_25K_015792", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=118.08, s=21.16, n=200, compute CI."}]}, "response": "95% CI ≈ [115.15, 121.01].", "meta": {"id": "WUA_STAT_25K_015793", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.04, μ=1.55, σ=22.7, compute z."}]}, "response": "z = 0.242.", "meta": {"id": "WUA_STAT_25K_015794", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.16, μ=89.27, σ=10.13, compute z."}]}, "response": "z = -1.689.", "meta": {"id": "WUA_STAT_25K_015795", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.45, compute E and Var."}]}, "response": "E[X]=12.60; Var[X]=6.930.", "meta": {"id": "WUA_STAT_25K_015796", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.58, μ=88.75, σ=14.47, compute z."}]}, "response": "z = -0.150.", "meta": {"id": "WUA_STAT_25K_015797", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.09, s=8.48, n=50, compute CI."}]}, "response": "95% CI ≈ [156.74, 161.44].", "meta": {"id": "WUA_STAT_25K_015798", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.019, sens=0.92, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.078.", "meta": {"id": "WUA_STAT_25K_015799", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.23, sens=0.95, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.876.", "meta": {"id": "WUA_STAT_25K_015800", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.46, compute E and Var."}]}, "response": "E[X]=10.58; Var[X]=5.713.", "meta": {"id": "WUA_STAT_25K_015801", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.116, sens=0.89, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.661.", "meta": {"id": "WUA_STAT_25K_015802", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.94, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.679.", "meta": {"id": "WUA_STAT_25K_015803", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.68, compute E and Var."}]}, "response": "E[X]=21.08; Var[X]=6.746.", "meta": {"id": "WUA_STAT_25K_015804", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=168.41, s=29.65, n=132, compute CI."}]}, "response": "95% CI ≈ [163.35, 173.47].", "meta": {"id": "WUA_STAT_25K_015805", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=36.94, μ=60.64, σ=23.77, compute z."}]}, "response": "z = -0.997.", "meta": {"id": "WUA_STAT_25K_015806", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.18, s=5.98, n=53, compute CI."}]}, "response": "95% CI ≈ [111.57, 114.79].", "meta": {"id": "WUA_STAT_25K_015807", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.58, s=34.66, n=104, compute CI."}]}, "response": "95% CI ≈ [180.92, 194.24].", "meta": {"id": "WUA_STAT_25K_015808", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.38, μ=42.17, σ=10.08, compute z."}]}, "response": "z = -0.178.", "meta": {"id": "WUA_STAT_25K_015809", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.146, sens=0.77, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.436.", "meta": {"id": "WUA_STAT_25K_015810", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.73, compute E and Var."}]}, "response": "E[X]=6.57; Var[X]=1.774.", "meta": {"id": "WUA_STAT_25K_015811", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=186.2, s=4.76, n=106, compute CI."}]}, "response": "95% CI ≈ [185.29, 187.11].", "meta": {"id": "WUA_STAT_25K_015812", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.154, sens=0.85, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.364.", "meta": {"id": "WUA_STAT_25K_015813", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.35, compute E and Var."}]}, "response": "E[X]=9.80; Var[X]=6.370.", "meta": {"id": "WUA_STAT_25K_015814", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.43, μ=94.02, σ=12.47, compute z."}]}, "response": "z = -6.944.", "meta": {"id": "WUA_STAT_25K_015815", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.099, sens=0.81, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.389.", "meta": {"id": "WUA_STAT_25K_015816", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.12, μ=32.56, σ=23.8, compute z."}]}, "response": "z = 1.326.", "meta": {"id": "WUA_STAT_25K_015817", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.54, μ=81.89, σ=12.09, compute z."}]}, "response": "z = -1.435.", "meta": {"id": "WUA_STAT_25K_015818", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.47, compute E and Var."}]}, "response": "E[X]=5.64; Var[X]=2.989.", "meta": {"id": "WUA_STAT_25K_015819", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.99, μ=84.67, σ=18.26, compute z."}]}, "response": "z = -3.323.", "meta": {"id": "WUA_STAT_25K_015820", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.56, μ=83.78, σ=22.94, compute z."}]}, "response": "z = 0.078.", "meta": {"id": "WUA_STAT_25K_015821", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.46, compute E and Var."}]}, "response": "E[X]=2.30; Var[X]=1.242.", "meta": {"id": "WUA_STAT_25K_015822", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.23, s=20.24, n=67, compute CI."}]}, "response": "95% CI ≈ [39.38, 49.08].", "meta": {"id": "WUA_STAT_25K_015823", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.107, sens=0.91, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.522.", "meta": {"id": "WUA_STAT_25K_015824", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.118, sens=0.97, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.590.", "meta": {"id": "WUA_STAT_25K_015825", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.73, compute E and Var."}]}, "response": "E[X]=26.28; Var[X]=7.096.", "meta": {"id": "WUA_STAT_25K_015826", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.99, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.314.", "meta": {"id": "WUA_STAT_25K_015827", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.11, compute E and Var."}]}, "response": "E[X]=3.08; Var[X]=2.741.", "meta": {"id": "WUA_STAT_25K_015828", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.34, s=24.76, n=35, compute CI."}]}, "response": "95% CI ≈ [155.14, 171.54].", "meta": {"id": "WUA_STAT_25K_015829", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.9, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.277.", "meta": {"id": "WUA_STAT_25K_015830", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.95, μ=10.32, σ=4.79, compute z."}]}, "response": "z = 6.186.", "meta": {"id": "WUA_STAT_25K_015831", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.032, sens=0.99, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.290.", "meta": {"id": "WUA_STAT_25K_015832", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.76, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.912.", "meta": {"id": "WUA_STAT_25K_015833", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.17, compute E and Var."}]}, "response": "E[X]=1.70; Var[X]=1.411.", "meta": {"id": "WUA_STAT_25K_015834", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.36, compute E and Var."}]}, "response": "E[X]=12.60; Var[X]=8.064.", "meta": {"id": "WUA_STAT_25K_015835", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=28.47, s=16.69, n=18, compute CI."}]}, "response": "95% CI ≈ [20.76, 36.18].", "meta": {"id": "WUA_STAT_25K_015836", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=193.0, s=11.86, n=79, compute CI."}]}, "response": "95% CI ≈ [190.38, 195.62].", "meta": {"id": "WUA_STAT_25K_015837", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.54, s=39.65, n=195, compute CI."}]}, "response": "95% CI ≈ [86.97, 98.11].", "meta": {"id": "WUA_STAT_25K_015838", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.47, μ=29.06, σ=6.45, compute z."}]}, "response": "z = 7.816.", "meta": {"id": "WUA_STAT_25K_015839", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.2, compute E and Var."}]}, "response": "E[X]=5.40; Var[X]=4.320.", "meta": {"id": "WUA_STAT_25K_015840", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.36, compute E and Var."}]}, "response": "E[X]=7.56; Var[X]=4.838.", "meta": {"id": "WUA_STAT_25K_015841", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.201, sens=0.9, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.850.", "meta": {"id": "WUA_STAT_25K_015842", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.75, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.106.", "meta": {"id": "WUA_STAT_25K_015843", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.08, μ=82.27, σ=18.21, compute z."}]}, "response": "z = -4.019.", "meta": {"id": "WUA_STAT_25K_015844", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.4, μ=49.44, σ=12.21, compute z."}]}, "response": "z = -1.314.", "meta": {"id": "WUA_STAT_25K_015845", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.03, μ=40.38, σ=7.47, compute z."}]}, "response": "z = -5.268.", "meta": {"id": "WUA_STAT_25K_015846", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.4, μ=81.97, σ=22.04, compute z."}]}, "response": "z = -2.612.", "meta": {"id": "WUA_STAT_25K_015847", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.34, μ=10.89, σ=22.72, compute z."}]}, "response": "z = -0.332.", "meta": {"id": "WUA_STAT_25K_015848", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.87, μ=73.81, σ=10.57, compute z."}]}, "response": "z = -3.779.", "meta": {"id": "WUA_STAT_25K_015849", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.43, μ=28.68, σ=18.65, compute z."}]}, "response": "z = -0.442.", "meta": {"id": "WUA_STAT_25K_015850", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.47, s=39.36, n=132, compute CI."}]}, "response": "95% CI ≈ [192.76, 206.18].", "meta": {"id": "WUA_STAT_25K_015851", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.97, μ=8.85, σ=19.49, compute z."}]}, "response": "z = 3.495.", "meta": {"id": "WUA_STAT_25K_015852", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=149.71, s=38.94, n=105, compute CI."}]}, "response": "95% CI ≈ [142.26, 157.16].", "meta": {"id": "WUA_STAT_25K_015853", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=112.99, s=28.62, n=181, compute CI."}]}, "response": "95% CI ≈ [108.82, 117.16].", "meta": {"id": "WUA_STAT_25K_015854", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.99, s=4.65, n=45, compute CI."}]}, "response": "95% CI ≈ [134.63, 137.35].", "meta": {"id": "WUA_STAT_25K_015855", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=164.04, s=25.89, n=118, compute CI."}]}, "response": "95% CI ≈ [159.37, 168.71].", "meta": {"id": "WUA_STAT_25K_015856", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.46, s=10.14, n=70, compute CI."}]}, "response": "95% CI ≈ [65.08, 69.84].", "meta": {"id": "WUA_STAT_25K_015857", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.26, μ=20.75, σ=3.7, compute z."}]}, "response": "z = 14.732.", "meta": {"id": "WUA_STAT_25K_015858", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.138, sens=0.72, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.390.", "meta": {"id": "WUA_STAT_25K_015859", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.39, μ=49.21, σ=15.9, compute z."}]}, "response": "z = 0.829.", "meta": {"id": "WUA_STAT_25K_015860", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.82, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.663.", "meta": {"id": "WUA_STAT_25K_015861", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.71, compute E and Var."}]}, "response": "E[X]=26.27; Var[X]=7.618.", "meta": {"id": "WUA_STAT_25K_015862", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.96, μ=71.78, σ=6.65, compute z."}]}, "response": "z = 3.185.", "meta": {"id": "WUA_STAT_25K_015863", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.49, compute E and Var."}]}, "response": "E[X]=3.92; Var[X]=1.999.", "meta": {"id": "WUA_STAT_25K_015864", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.15, compute E and Var."}]}, "response": "E[X]=3.45; Var[X]=2.932.", "meta": {"id": "WUA_STAT_25K_015865", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.106, sens=0.72, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.262.", "meta": {"id": "WUA_STAT_25K_015866", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.32, s=5.73, n=128, compute CI."}]}, "response": "95% CI ≈ [138.33, 140.31].", "meta": {"id": "WUA_STAT_25K_015867", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.097, sens=0.92, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.712.", "meta": {"id": "WUA_STAT_25K_015868", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.94, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.479.", "meta": {"id": "WUA_STAT_25K_015869", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.029, sens=0.73, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.165.", "meta": {"id": "WUA_STAT_25K_015870", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.09, sens=0.86, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.233.", "meta": {"id": "WUA_STAT_25K_015871", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.58, μ=43.76, σ=5.0, compute z."}]}, "response": "z = 4.964.", "meta": {"id": "WUA_STAT_25K_015872", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.54, compute E and Var."}]}, "response": "E[X]=7.56; Var[X]=3.478.", "meta": {"id": "WUA_STAT_25K_015873", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.66, μ=81.27, σ=5.16, compute z."}]}, "response": "z = 1.820.", "meta": {"id": "WUA_STAT_25K_015874", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.65, compute E and Var."}]}, "response": "E[X]=5.20; Var[X]=1.820.", "meta": {"id": "WUA_STAT_25K_015875", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.65, s=18.31, n=84, compute CI."}]}, "response": "95% CI ≈ [105.73, 113.57].", "meta": {"id": "WUA_STAT_25K_015876", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.197, sens=0.71, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.521.", "meta": {"id": "WUA_STAT_25K_015877", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.21, sens=0.94, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.463.", "meta": {"id": "WUA_STAT_25K_015878", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.99, μ=36.23, σ=18.53, compute z."}]}, "response": "z = 3.171.", "meta": {"id": "WUA_STAT_25K_015879", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.014, sens=0.95, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.055.", "meta": {"id": "WUA_STAT_25K_015880", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.11, μ=46.11, σ=24.28, compute z."}]}, "response": "z = -1.442.", "meta": {"id": "WUA_STAT_25K_015881", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.66, s=34.57, n=152, compute CI."}]}, "response": "95% CI ≈ [122.16, 133.16].", "meta": {"id": "WUA_STAT_25K_015882", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.18, μ=87.76, σ=9.37, compute z."}]}, "response": "z = -5.291.", "meta": {"id": "WUA_STAT_25K_015883", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.99, μ=90.59, σ=12.16, compute z."}]}, "response": "z = -4.737.", "meta": {"id": "WUA_STAT_25K_015884", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.73, compute E and Var."}]}, "response": "E[X]=24.09; Var[X]=6.504.", "meta": {"id": "WUA_STAT_25K_015885", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.67, μ=0.77, σ=1.73, compute z."}]}, "response": "z = 20.173.", "meta": {"id": "WUA_STAT_25K_015886", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.95, s=39.57, n=136, compute CI."}]}, "response": "95% CI ≈ [172.30, 185.60].", "meta": {"id": "WUA_STAT_25K_015887", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.0, s=21.86, n=76, compute CI."}]}, "response": "95% CI ≈ [24.09, 33.91].", "meta": {"id": "WUA_STAT_25K_015888", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.42, compute E and Var."}]}, "response": "E[X]=13.86; Var[X]=8.039.", "meta": {"id": "WUA_STAT_25K_015889", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.6, μ=83.5, σ=20.27, compute z."}]}, "response": "z = -1.179.", "meta": {"id": "WUA_STAT_25K_015890", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.217, sens=0.73, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.574.", "meta": {"id": "WUA_STAT_25K_015891", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.0, s=29.26, n=36, compute CI."}]}, "response": "95% CI ≈ [141.44, 160.56].", "meta": {"id": "WUA_STAT_25K_015892", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.111, sens=0.73, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.646.", "meta": {"id": "WUA_STAT_25K_015893", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.64, μ=82.58, σ=2.21, compute z."}]}, "response": "z = -24.860.", "meta": {"id": "WUA_STAT_25K_015894", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=104.93, s=14.2, n=99, compute CI."}]}, "response": "95% CI ≈ [102.13, 107.73].", "meta": {"id": "WUA_STAT_25K_015895", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.08, compute E and Var."}]}, "response": "E[X]=2.08; Var[X]=1.914.", "meta": {"id": "WUA_STAT_25K_015896", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.18, μ=86.13, σ=19.4, compute z."}]}, "response": "z = -2.265.", "meta": {"id": "WUA_STAT_25K_015897", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.58, compute E and Var."}]}, "response": "E[X]=9.28; Var[X]=3.898.", "meta": {"id": "WUA_STAT_25K_015898", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.91, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.457.", "meta": {"id": "WUA_STAT_25K_015899", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.11, μ=62.48, σ=12.21, compute z."}]}, "response": "z = -4.535.", "meta": {"id": "WUA_STAT_25K_015900", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.038, sens=0.88, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.170.", "meta": {"id": "WUA_STAT_25K_015901", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.12, sens=0.95, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.371.", "meta": {"id": "WUA_STAT_25K_015902", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.13, s=31.33, n=182, compute CI."}]}, "response": "95% CI ≈ [26.58, 35.68].", "meta": {"id": "WUA_STAT_25K_015903", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.66, s=37.81, n=107, compute CI."}]}, "response": "95% CI ≈ [23.50, 37.82].", "meta": {"id": "WUA_STAT_25K_015904", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.53, s=6.89, n=96, compute CI."}]}, "response": "95% CI ≈ [43.15, 45.91].", "meta": {"id": "WUA_STAT_25K_015905", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.93, μ=0.04, σ=14.27, compute z."}]}, "response": "z = 0.553.", "meta": {"id": "WUA_STAT_25K_015906", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.88, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.256.", "meta": {"id": "WUA_STAT_25K_015907", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.28, s=23.45, n=139, compute CI."}]}, "response": "95% CI ≈ [155.38, 163.18].", "meta": {"id": "WUA_STAT_25K_015908", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.11, compute E and Var."}]}, "response": "E[X]=1.43; Var[X]=1.273.", "meta": {"id": "WUA_STAT_25K_015909", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.33, compute E and Var."}]}, "response": "E[X]=5.28; Var[X]=3.538.", "meta": {"id": "WUA_STAT_25K_015910", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=10.52, μ=98.69, σ=14.78, compute z."}]}, "response": "z = -5.965.", "meta": {"id": "WUA_STAT_25K_015911", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=93.73, μ=47.9, σ=20.2, compute z."}]}, "response": "z = 2.269.", "meta": {"id": "WUA_STAT_25K_015912", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.58, μ=87.37, σ=1.68, compute z."}]}, "response": "z = -33.804.", "meta": {"id": "WUA_STAT_25K_015913", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.55, μ=14.22, σ=17.95, compute z."}]}, "response": "z = 4.085.", "meta": {"id": "WUA_STAT_25K_015914", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.12, compute E and Var."}]}, "response": "E[X]=1.92; Var[X]=1.690.", "meta": {"id": "WUA_STAT_25K_015915", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.98, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.440.", "meta": {"id": "WUA_STAT_25K_015916", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.25, μ=2.93, σ=10.41, compute z."}]}, "response": "z = 5.698.", "meta": {"id": "WUA_STAT_25K_015917", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=71.82, s=22.52, n=171, compute CI."}]}, "response": "95% CI ≈ [68.44, 75.20].", "meta": {"id": "WUA_STAT_25K_015918", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=96.86, s=29.65, n=56, compute CI."}]}, "response": "95% CI ≈ [89.09, 104.63].", "meta": {"id": "WUA_STAT_25K_015919", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.52, compute E and Var."}]}, "response": "E[X]=8.32; Var[X]=3.994.", "meta": {"id": "WUA_STAT_25K_015920", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.93, s=8.05, n=181, compute CI."}]}, "response": "95% CI ≈ [129.76, 132.10].", "meta": {"id": "WUA_STAT_25K_015921", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.06, compute E and Var."}]}, "response": "E[X]=2.22; Var[X]=2.087.", "meta": {"id": "WUA_STAT_25K_015922", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.31, μ=26.1, σ=4.44, compute z."}]}, "response": "z = 7.255.", "meta": {"id": "WUA_STAT_25K_015923", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.74, compute E and Var."}]}, "response": "E[X]=28.86; Var[X]=7.504.", "meta": {"id": "WUA_STAT_25K_015924", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.76, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.473.", "meta": {"id": "WUA_STAT_25K_015925", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.34, μ=34.42, σ=11.59, compute z."}]}, "response": "z = 0.338.", "meta": {"id": "WUA_STAT_25K_015926", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=116.51, s=29.63, n=78, compute CI."}]}, "response": "95% CI ≈ [109.93, 123.09].", "meta": {"id": "WUA_STAT_25K_015927", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.28, compute E and Var."}]}, "response": "E[X]=8.12; Var[X]=5.846.", "meta": {"id": "WUA_STAT_25K_015928", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.25, μ=93.86, σ=17.87, compute z."}]}, "response": "z = -2.217.", "meta": {"id": "WUA_STAT_25K_015929", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.66, μ=71.68, σ=24.83, compute z."}]}, "response": "z = 1.046.", "meta": {"id": "WUA_STAT_25K_015930", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.96, μ=18.95, σ=17.85, compute z."}]}, "response": "z = 2.634.", "meta": {"id": "WUA_STAT_25K_015931", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.08, s=14.8, n=186, compute CI."}]}, "response": "95% CI ≈ [51.95, 56.21].", "meta": {"id": "WUA_STAT_25K_015932", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.05, compute E and Var."}]}, "response": "E[X]=1.70; Var[X]=1.615.", "meta": {"id": "WUA_STAT_25K_015933", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.61, compute E and Var."}]}, "response": "E[X]=10.98; Var[X]=4.282.", "meta": {"id": "WUA_STAT_25K_015934", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.23, compute E and Var."}]}, "response": "E[X]=3.91; Var[X]=3.011.", "meta": {"id": "WUA_STAT_25K_015935", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.205, sens=0.88, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.883.", "meta": {"id": "WUA_STAT_25K_015936", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.16, compute E and Var."}]}, "response": "E[X]=3.04; Var[X]=2.554.", "meta": {"id": "WUA_STAT_25K_015937", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.86, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.424.", "meta": {"id": "WUA_STAT_25K_015938", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.193, sens=0.95, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.558.", "meta": {"id": "WUA_STAT_25K_015939", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.165, sens=0.87, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.711.", "meta": {"id": "WUA_STAT_25K_015940", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.241, sens=0.93, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.937.", "meta": {"id": "WUA_STAT_25K_015941", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.37, compute E and Var."}]}, "response": "E[X]=14.80; Var[X]=9.324.", "meta": {"id": "WUA_STAT_25K_015942", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.67, compute E and Var."}]}, "response": "E[X]=5.36; Var[X]=1.769.", "meta": {"id": "WUA_STAT_25K_015943", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.039, sens=0.76, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.096.", "meta": {"id": "WUA_STAT_25K_015944", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.14, s=35.44, n=194, compute CI."}]}, "response": "95% CI ≈ [133.15, 143.13].", "meta": {"id": "WUA_STAT_25K_015945", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.38, compute E and Var."}]}, "response": "E[X]=7.98; Var[X]=4.948.", "meta": {"id": "WUA_STAT_25K_015946", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=159.51, s=24.37, n=125, compute CI."}]}, "response": "95% CI ≈ [155.24, 163.78].", "meta": {"id": "WUA_STAT_25K_015947", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.81, s=1.99, n=115, compute CI."}]}, "response": "95% CI ≈ [157.45, 158.17].", "meta": {"id": "WUA_STAT_25K_015948", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.98, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.400.", "meta": {"id": "WUA_STAT_25K_015949", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.16, s=32.13, n=189, compute CI."}]}, "response": "95% CI ≈ [177.58, 186.74].", "meta": {"id": "WUA_STAT_25K_015950", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.03, s=30.87, n=26, compute CI."}]}, "response": "95% CI ≈ [138.16, 161.90].", "meta": {"id": "WUA_STAT_25K_015951", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.16, compute E and Var."}]}, "response": "E[X]=5.92; Var[X]=4.973.", "meta": {"id": "WUA_STAT_25K_015952", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.74, μ=90.93, σ=22.3, compute z."}]}, "response": "z = -2.116.", "meta": {"id": "WUA_STAT_25K_015953", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.52, compute E and Var."}]}, "response": "E[X]=17.68; Var[X]=8.486.", "meta": {"id": "WUA_STAT_25K_015954", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.18, s=18.35, n=164, compute CI."}]}, "response": "95% CI ≈ [75.37, 80.99].", "meta": {"id": "WUA_STAT_25K_015955", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.13, compute E and Var."}]}, "response": "E[X]=2.99; Var[X]=2.601.", "meta": {"id": "WUA_STAT_25K_015956", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.61, compute E and Var."}]}, "response": "E[X]=22.57; Var[X]=8.802.", "meta": {"id": "WUA_STAT_25K_015957", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.09, s=4.31, n=158, compute CI."}]}, "response": "95% CI ≈ [61.42, 62.76].", "meta": {"id": "WUA_STAT_25K_015958", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.96, s=36.52, n=143, compute CI."}]}, "response": "95% CI ≈ [34.97, 46.95].", "meta": {"id": "WUA_STAT_25K_015959", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.98, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.729.", "meta": {"id": "WUA_STAT_25K_015960", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.125, sens=0.72, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.340.", "meta": {"id": "WUA_STAT_25K_015961", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=58.87, s=9.08, n=90, compute CI."}]}, "response": "95% CI ≈ [56.99, 60.75].", "meta": {"id": "WUA_STAT_25K_015962", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=149.91, s=37.54, n=66, compute CI."}]}, "response": "95% CI ≈ [140.85, 158.97].", "meta": {"id": "WUA_STAT_25K_015963", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.63, compute E and Var."}]}, "response": "E[X]=18.90; Var[X]=6.993.", "meta": {"id": "WUA_STAT_25K_015964", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.42, s=18.77, n=154, compute CI."}]}, "response": "95% CI ≈ [119.46, 125.38].", "meta": {"id": "WUA_STAT_25K_015965", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.97, μ=1.06, σ=3.23, compute z."}]}, "response": "z = 3.997.", "meta": {"id": "WUA_STAT_25K_015966", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.57, μ=79.95, σ=15.49, compute z."}]}, "response": "z = -1.768.", "meta": {"id": "WUA_STAT_25K_015967", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.67, s=12.08, n=180, compute CI."}]}, "response": "95% CI ≈ [38.91, 42.43].", "meta": {"id": "WUA_STAT_25K_015968", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.05, s=36.98, n=53, compute CI."}]}, "response": "95% CI ≈ [146.09, 166.01].", "meta": {"id": "WUA_STAT_25K_015969", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.88, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.564.", "meta": {"id": "WUA_STAT_25K_015970", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.06, compute E and Var."}]}, "response": "E[X]=1.56; Var[X]=1.466.", "meta": {"id": "WUA_STAT_25K_015971", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.43, μ=39.09, σ=3.25, compute z."}]}, "response": "z = -0.203.", "meta": {"id": "WUA_STAT_25K_015972", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.05, compute E and Var."}]}, "response": "E[X]=0.70; Var[X]=0.665.", "meta": {"id": "WUA_STAT_25K_015973", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.39, s=7.76, n=114, compute CI."}]}, "response": "95% CI ≈ [44.97, 47.81].", "meta": {"id": "WUA_STAT_25K_015974", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.9, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.851.", "meta": {"id": "WUA_STAT_25K_015975", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.74, s=8.58, n=112, compute CI."}]}, "response": "95% CI ≈ [98.15, 101.33].", "meta": {"id": "WUA_STAT_25K_015976", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.72, s=33.38, n=123, compute CI."}]}, "response": "95% CI ≈ [54.82, 66.62].", "meta": {"id": "WUA_STAT_25K_015977", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.22, compute E and Var."}]}, "response": "E[X]=3.96; Var[X]=3.089.", "meta": {"id": "WUA_STAT_25K_015978", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.3, compute E and Var."}]}, "response": "E[X]=11.40; Var[X]=7.980.", "meta": {"id": "WUA_STAT_25K_015979", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.75, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.051.", "meta": {"id": "WUA_STAT_25K_015980", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.243, sens=0.94, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.519.", "meta": {"id": "WUA_STAT_25K_015981", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=34.1, s=9.4, n=93, compute CI."}]}, "response": "95% CI ≈ [32.19, 36.01].", "meta": {"id": "WUA_STAT_25K_015982", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.064, sens=0.9, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.266.", "meta": {"id": "WUA_STAT_25K_015983", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.22, s=31.98, n=192, compute CI."}]}, "response": "95% CI ≈ [124.70, 133.74].", "meta": {"id": "WUA_STAT_25K_015984", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.43, s=27.6, n=110, compute CI."}]}, "response": "95% CI ≈ [151.27, 161.59].", "meta": {"id": "WUA_STAT_25K_015985", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=70.21, μ=31.88, σ=7.43, compute z."}]}, "response": "z = 5.159.", "meta": {"id": "WUA_STAT_25K_015986", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=93.97, s=21.96, n=126, compute CI."}]}, "response": "95% CI ≈ [90.14, 97.80].", "meta": {"id": "WUA_STAT_25K_015987", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.6, μ=31.38, σ=14.02, compute z."}]}, "response": "z = 2.441.", "meta": {"id": "WUA_STAT_25K_015988", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=61.82, s=29.14, n=18, compute CI."}]}, "response": "95% CI ≈ [48.36, 75.28].", "meta": {"id": "WUA_STAT_25K_015989", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.86, μ=86.88, σ=7.39, compute z."}]}, "response": "z = -3.386.", "meta": {"id": "WUA_STAT_25K_015990", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=113.74, s=29.0, n=52, compute CI."}]}, "response": "95% CI ≈ [105.86, 121.62].", "meta": {"id": "WUA_STAT_25K_015991", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.38, s=5.35, n=38, compute CI."}]}, "response": "95% CI ≈ [123.68, 127.08].", "meta": {"id": "WUA_STAT_25K_015992", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.14, μ=91.57, σ=10.33, compute z."}]}, "response": "z = -2.946.", "meta": {"id": "WUA_STAT_25K_015993", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.66, μ=95.09, σ=13.19, compute z."}]}, "response": "z = -3.975.", "meta": {"id": "WUA_STAT_25K_015994", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.12, sens=0.82, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.309.", "meta": {"id": "WUA_STAT_25K_015995", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.247, sens=0.79, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.481.", "meta": {"id": "WUA_STAT_25K_015996", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.83, μ=28.7, σ=9.93, compute z."}]}, "response": "z = 4.847.", "meta": {"id": "WUA_STAT_25K_015997", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.93, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.674.", "meta": {"id": "WUA_STAT_25K_015998", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=54.92, s=20.78, n=45, compute CI."}]}, "response": "95% CI ≈ [48.85, 60.99].", "meta": {"id": "WUA_STAT_25K_015999", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.85, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.327.", "meta": {"id": "WUA_STAT_25K_016000", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.63, μ=17.59, σ=5.82, compute z."}]}, "response": "z = 10.832.", "meta": {"id": "WUA_STAT_25K_016001", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=148.19, s=26.72, n=15, compute CI."}]}, "response": "95% CI ≈ [134.67, 161.71].", "meta": {"id": "WUA_STAT_25K_016002", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.44, compute E and Var."}]}, "response": "E[X]=17.16; Var[X]=9.610.", "meta": {"id": "WUA_STAT_25K_016003", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=139.44, s=5.6, n=200, compute CI."}]}, "response": "95% CI ≈ [138.66, 140.22].", "meta": {"id": "WUA_STAT_25K_016004", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.55, μ=34.26, σ=4.29, compute z."}]}, "response": "z = -1.331.", "meta": {"id": "WUA_STAT_25K_016005", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=56.92, s=5.05, n=198, compute CI."}]}, "response": "95% CI ≈ [56.22, 57.62].", "meta": {"id": "WUA_STAT_25K_016006", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.18, s=11.0, n=168, compute CI."}]}, "response": "95% CI ≈ [42.52, 45.84].", "meta": {"id": "WUA_STAT_25K_016007", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.36, s=8.85, n=15, compute CI."}]}, "response": "95% CI ≈ [38.88, 47.84].", "meta": {"id": "WUA_STAT_25K_016008", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=66.98, s=31.28, n=78, compute CI."}]}, "response": "95% CI ≈ [60.04, 73.92].", "meta": {"id": "WUA_STAT_25K_016009", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.7, compute E and Var."}]}, "response": "E[X]=4.90; Var[X]=1.470.", "meta": {"id": "WUA_STAT_25K_016010", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.11, s=1.49, n=28, compute CI."}]}, "response": "95% CI ≈ [42.56, 43.66].", "meta": {"id": "WUA_STAT_25K_016011", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.98, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.712.", "meta": {"id": "WUA_STAT_25K_016012", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=181.58, s=32.58, n=94, compute CI."}]}, "response": "95% CI ≈ [174.99, 188.17].", "meta": {"id": "WUA_STAT_25K_016013", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.55, compute E and Var."}]}, "response": "E[X]=14.30; Var[X]=6.435.", "meta": {"id": "WUA_STAT_25K_016014", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.61, compute E and Var."}]}, "response": "E[X]=7.93; Var[X]=3.093.", "meta": {"id": "WUA_STAT_25K_016015", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.33, μ=77.55, σ=14.6, compute z."}]}, "response": "z = -5.221.", "meta": {"id": "WUA_STAT_25K_016016", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.93, s=18.96, n=151, compute CI."}]}, "response": "95% CI ≈ [26.91, 32.95].", "meta": {"id": "WUA_STAT_25K_016017", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.16, compute E and Var."}]}, "response": "E[X]=3.52; Var[X]=2.957.", "meta": {"id": "WUA_STAT_25K_016018", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.94, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.630.", "meta": {"id": "WUA_STAT_25K_016019", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.7, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=2.520.", "meta": {"id": "WUA_STAT_25K_016020", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=122.31, s=31.65, n=199, compute CI."}]}, "response": "95% CI ≈ [117.91, 126.71].", "meta": {"id": "WUA_STAT_25K_016021", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.13, compute E and Var."}]}, "response": "E[X]=2.34; Var[X]=2.036.", "meta": {"id": "WUA_STAT_25K_016022", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.43, s=31.08, n=179, compute CI."}]}, "response": "95% CI ≈ [75.88, 84.98].", "meta": {"id": "WUA_STAT_25K_016023", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.65, μ=82.94, σ=24.92, compute z."}]}, "response": "z = -2.941.", "meta": {"id": "WUA_STAT_25K_016024", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.54, μ=62.83, σ=9.44, compute z."}]}, "response": "z = -3.844.", "meta": {"id": "WUA_STAT_25K_016025", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.4, s=38.32, n=38, compute CI."}]}, "response": "95% CI ≈ [28.22, 52.58].", "meta": {"id": "WUA_STAT_25K_016026", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.051, sens=0.76, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.177.", "meta": {"id": "WUA_STAT_25K_016027", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.98, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.846.", "meta": {"id": "WUA_STAT_25K_016028", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.7, s=23.2, n=140, compute CI."}]}, "response": "95% CI ≈ [174.86, 182.54].", "meta": {"id": "WUA_STAT_25K_016029", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.66, μ=24.47, σ=6.61, compute z."}]}, "response": "z = -2.846.", "meta": {"id": "WUA_STAT_25K_016030", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.66, compute E and Var."}]}, "response": "E[X]=10.56; Var[X]=3.590.", "meta": {"id": "WUA_STAT_25K_016031", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=28.18, s=38.95, n=31, compute CI."}]}, "response": "95% CI ≈ [14.47, 41.89].", "meta": {"id": "WUA_STAT_25K_016032", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=76.19, s=20.85, n=81, compute CI."}]}, "response": "95% CI ≈ [71.65, 80.73].", "meta": {"id": "WUA_STAT_25K_016033", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.14, μ=7.65, σ=8.03, compute z."}]}, "response": "z = 0.559.", "meta": {"id": "WUA_STAT_25K_016034", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.8, μ=1.47, σ=7.43, compute z."}]}, "response": "z = 1.794.", "meta": {"id": "WUA_STAT_25K_016035", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.99, μ=72.32, σ=23.43, compute z."}]}, "response": "z = -2.447.", "meta": {"id": "WUA_STAT_25K_016036", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.59, compute E and Var."}]}, "response": "E[X]=21.24; Var[X]=8.708.", "meta": {"id": "WUA_STAT_25K_016037", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.25, compute E and Var."}]}, "response": "E[X]=9.25; Var[X]=6.938.", "meta": {"id": "WUA_STAT_25K_016038", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.74, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.045.", "meta": {"id": "WUA_STAT_25K_016039", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.3, μ=77.51, σ=4.32, compute z."}]}, "response": "z = -10.697.", "meta": {"id": "WUA_STAT_25K_016040", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.15, compute E and Var."}]}, "response": "E[X]=0.75; Var[X]=0.637.", "meta": {"id": "WUA_STAT_25K_016041", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.016, sens=0.98, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.109.", "meta": {"id": "WUA_STAT_25K_016042", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.75, s=23.68, n=49, compute CI."}]}, "response": "95% CI ≈ [39.12, 52.38].", "meta": {"id": "WUA_STAT_25K_016043", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.66, compute E and Var."}]}, "response": "E[X]=17.82; Var[X]=6.059.", "meta": {"id": "WUA_STAT_25K_016044", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.28, μ=38.1, σ=8.81, compute z."}]}, "response": "z = 0.815.", "meta": {"id": "WUA_STAT_25K_016045", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.46, s=31.44, n=82, compute CI."}]}, "response": "95% CI ≈ [90.65, 104.27].", "meta": {"id": "WUA_STAT_25K_016046", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.64, compute E and Var."}]}, "response": "E[X]=10.24; Var[X]=3.686.", "meta": {"id": "WUA_STAT_25K_016047", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.65, s=39.8, n=76, compute CI."}]}, "response": "95% CI ≈ [121.70, 139.60].", "meta": {"id": "WUA_STAT_25K_016048", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=65.03, μ=54.85, σ=13.5, compute z."}]}, "response": "z = 0.754.", "meta": {"id": "WUA_STAT_25K_016049", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.173, sens=0.97, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.610.", "meta": {"id": "WUA_STAT_25K_016050", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.14, sens=0.74, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.388.", "meta": {"id": "WUA_STAT_25K_016051", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.72, compute E and Var."}]}, "response": "E[X]=21.60; Var[X]=6.048.", "meta": {"id": "WUA_STAT_25K_016052", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.35, compute E and Var."}]}, "response": "E[X]=6.65; Var[X]=4.322.", "meta": {"id": "WUA_STAT_25K_016053", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.41, s=33.26, n=183, compute CI."}]}, "response": "95% CI ≈ [79.59, 89.23].", "meta": {"id": "WUA_STAT_25K_016054", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.97, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.166.", "meta": {"id": "WUA_STAT_25K_016055", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.5, s=26.48, n=169, compute CI."}]}, "response": "95% CI ≈ [141.51, 149.49].", "meta": {"id": "WUA_STAT_25K_016056", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.15, s=17.96, n=13, compute CI."}]}, "response": "95% CI ≈ [67.39, 86.91].", "meta": {"id": "WUA_STAT_25K_016057", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.027, sens=0.92, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.561.", "meta": {"id": "WUA_STAT_25K_016058", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.92, μ=9.37, σ=22.73, compute z."}]}, "response": "z = 2.092.", "meta": {"id": "WUA_STAT_25K_016059", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=41.52, s=20.7, n=180, compute CI."}]}, "response": "95% CI ≈ [38.50, 44.54].", "meta": {"id": "WUA_STAT_25K_016060", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.84, μ=52.12, σ=21.38, compute z."}]}, "response": "z = -0.060.", "meta": {"id": "WUA_STAT_25K_016061", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.28, μ=16.82, σ=4.99, compute z."}]}, "response": "z = 2.697.", "meta": {"id": "WUA_STAT_25K_016062", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.185, sens=0.87, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.497.", "meta": {"id": "WUA_STAT_25K_016063", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.99, s=38.61, n=168, compute CI."}]}, "response": "95% CI ≈ [185.15, 196.83].", "meta": {"id": "WUA_STAT_25K_016064", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=115.35, s=12.87, n=23, compute CI."}]}, "response": "95% CI ≈ [110.09, 120.61].", "meta": {"id": "WUA_STAT_25K_016065", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.73, compute E and Var."}]}, "response": "E[X]=7.30; Var[X]=1.971.", "meta": {"id": "WUA_STAT_25K_016066", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=149.95, s=13.75, n=58, compute CI."}]}, "response": "95% CI ≈ [146.41, 153.49].", "meta": {"id": "WUA_STAT_25K_016067", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.22, μ=61.66, σ=18.28, compute z."}]}, "response": "z = -2.322.", "meta": {"id": "WUA_STAT_25K_016068", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.9, μ=95.39, σ=1.83, compute z."}]}, "response": "z = -12.836.", "meta": {"id": "WUA_STAT_25K_016069", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.34, compute E and Var."}]}, "response": "E[X]=7.48; Var[X]=4.937.", "meta": {"id": "WUA_STAT_25K_016070", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.68, compute E and Var."}]}, "response": "E[X]=18.36; Var[X]=5.875.", "meta": {"id": "WUA_STAT_25K_016071", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.59, s=39.18, n=52, compute CI."}]}, "response": "95% CI ≈ [126.94, 148.24].", "meta": {"id": "WUA_STAT_25K_016072", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.59, s=4.99, n=184, compute CI."}]}, "response": "95% CI ≈ [28.87, 30.31].", "meta": {"id": "WUA_STAT_25K_016073", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.47, compute E and Var."}]}, "response": "E[X]=18.33; Var[X]=9.715.", "meta": {"id": "WUA_STAT_25K_016074", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.61, μ=54.36, σ=3.83, compute z."}]}, "response": "z = -3.068.", "meta": {"id": "WUA_STAT_25K_016075", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.96, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.396.", "meta": {"id": "WUA_STAT_25K_016076", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.01, μ=82.06, σ=6.94, compute z."}]}, "response": "z = 2.154.", "meta": {"id": "WUA_STAT_25K_016077", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=84.57, s=34.77, n=165, compute CI."}]}, "response": "95% CI ≈ [79.26, 89.88].", "meta": {"id": "WUA_STAT_25K_016078", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.19, compute E and Var."}]}, "response": "E[X]=6.84; Var[X]=5.540.", "meta": {"id": "WUA_STAT_25K_016079", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.182, sens=0.77, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.417.", "meta": {"id": "WUA_STAT_25K_016080", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.79, compute E and Var."}]}, "response": "E[X]=24.49; Var[X]=5.143.", "meta": {"id": "WUA_STAT_25K_016081", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.77, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.248.", "meta": {"id": "WUA_STAT_25K_016082", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.44, compute E and Var."}]}, "response": "E[X]=10.12; Var[X]=5.667.", "meta": {"id": "WUA_STAT_25K_016083", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.34, μ=81.41, σ=17.81, compute z."}]}, "response": "z = -3.261.", "meta": {"id": "WUA_STAT_25K_016084", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.205, sens=0.84, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.915.", "meta": {"id": "WUA_STAT_25K_016085", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.96, μ=92.98, σ=3.47, compute z."}]}, "response": "z = -14.991.", "meta": {"id": "WUA_STAT_25K_016086", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.74, compute E and Var."}]}, "response": "E[X]=26.64; Var[X]=6.926.", "meta": {"id": "WUA_STAT_25K_016087", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.56, compute E and Var."}]}, "response": "E[X]=13.44; Var[X]=5.914.", "meta": {"id": "WUA_STAT_25K_016088", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.39, s=25.12, n=136, compute CI."}]}, "response": "95% CI ≈ [71.17, 79.61].", "meta": {"id": "WUA_STAT_25K_016089", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.11, μ=61.72, σ=1.78, compute z."}]}, "response": "z = 10.893.", "meta": {"id": "WUA_STAT_25K_016090", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.031, sens=0.87, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.112.", "meta": {"id": "WUA_STAT_25K_016091", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.74, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.357.", "meta": {"id": "WUA_STAT_25K_016092", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.19, μ=71.75, σ=4.96, compute z."}]}, "response": "z = -13.218.", "meta": {"id": "WUA_STAT_25K_016093", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.11, s=10.7, n=133, compute CI."}]}, "response": "95% CI ≈ [182.29, 185.93].", "meta": {"id": "WUA_STAT_25K_016094", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.55, μ=93.61, σ=8.23, compute z."}]}, "response": "z = -5.597.", "meta": {"id": "WUA_STAT_25K_016095", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.025, sens=0.79, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.403.", "meta": {"id": "WUA_STAT_25K_016096", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.248, sens=0.91, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.750.", "meta": {"id": "WUA_STAT_25K_016097", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=104.86, s=15.77, n=68, compute CI."}]}, "response": "95% CI ≈ [101.11, 108.61].", "meta": {"id": "WUA_STAT_25K_016098", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.12, s=12.84, n=161, compute CI."}]}, "response": "95% CI ≈ [186.14, 190.10].", "meta": {"id": "WUA_STAT_25K_016099", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.34, compute E and Var."}]}, "response": "E[X]=11.22; Var[X]=7.405.", "meta": {"id": "WUA_STAT_25K_016100", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.94, μ=39.05, σ=16.96, compute z."}]}, "response": "z = 1.703.", "meta": {"id": "WUA_STAT_25K_016101", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.27, s=18.42, n=130, compute CI."}]}, "response": "95% CI ≈ [107.10, 113.44].", "meta": {"id": "WUA_STAT_25K_016102", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.82, μ=68.4, σ=23.88, compute z."}]}, "response": "z = -1.574.", "meta": {"id": "WUA_STAT_25K_016103", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=195.1, s=8.09, n=25, compute CI."}]}, "response": "95% CI ≈ [191.93, 198.27].", "meta": {"id": "WUA_STAT_25K_016104", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.83, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.664.", "meta": {"id": "WUA_STAT_25K_016105", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.63, μ=16.92, σ=19.67, compute z."}]}, "response": "z = -0.167.", "meta": {"id": "WUA_STAT_25K_016106", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.226, sens=0.97, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.563.", "meta": {"id": "WUA_STAT_25K_016107", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.74, μ=98.41, σ=13.76, compute z."}]}, "response": "z = -1.575.", "meta": {"id": "WUA_STAT_25K_016108", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.18, sens=0.9, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.452.", "meta": {"id": "WUA_STAT_25K_016109", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.99, μ=20.38, σ=3.67, compute z."}]}, "response": "z = 7.796.", "meta": {"id": "WUA_STAT_25K_016110", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.03, sens=0.74, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.186.", "meta": {"id": "WUA_STAT_25K_016111", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.51, μ=27.91, σ=18.97, compute z."}]}, "response": "z = -0.390.", "meta": {"id": "WUA_STAT_25K_016112", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.13, compute E and Var."}]}, "response": "E[X]=2.34; Var[X]=2.036.", "meta": {"id": "WUA_STAT_25K_016113", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=152.58, s=21.73, n=85, compute CI."}]}, "response": "95% CI ≈ [147.96, 157.20].", "meta": {"id": "WUA_STAT_25K_016114", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.144, sens=0.74, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.384.", "meta": {"id": "WUA_STAT_25K_016115", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=56.04, s=22.42, n=55, compute CI."}]}, "response": "95% CI ≈ [50.11, 61.97].", "meta": {"id": "WUA_STAT_25K_016116", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=143.69, s=30.69, n=55, compute CI."}]}, "response": "95% CI ≈ [135.58, 151.80].", "meta": {"id": "WUA_STAT_25K_016117", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.42, μ=3.0, σ=5.41, compute z."}]}, "response": "z = 3.959.", "meta": {"id": "WUA_STAT_25K_016118", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.033, sens=0.96, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.201.", "meta": {"id": "WUA_STAT_25K_016119", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.47, s=38.29, n=173, compute CI."}]}, "response": "95% CI ≈ [27.76, 39.18].", "meta": {"id": "WUA_STAT_25K_016120", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.78, compute E and Var."}]}, "response": "E[X]=25.74; Var[X]=5.663.", "meta": {"id": "WUA_STAT_25K_016121", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.04, μ=44.91, σ=11.74, compute z."}]}, "response": "z = 3.503.", "meta": {"id": "WUA_STAT_25K_016122", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.48, compute E and Var."}]}, "response": "E[X]=18.24; Var[X]=9.485.", "meta": {"id": "WUA_STAT_25K_016123", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.53, compute E and Var."}]}, "response": "E[X]=3.18; Var[X]=1.495.", "meta": {"id": "WUA_STAT_25K_016124", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.7, μ=27.36, σ=13.95, compute z."}]}, "response": "z = 0.239.", "meta": {"id": "WUA_STAT_25K_016125", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=56.64, s=30.58, n=121, compute CI."}]}, "response": "95% CI ≈ [51.19, 62.09].", "meta": {"id": "WUA_STAT_25K_016126", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.83, μ=68.42, σ=22.66, compute z."}]}, "response": "z = -1.703.", "meta": {"id": "WUA_STAT_25K_016127", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.63, s=21.13, n=102, compute CI."}]}, "response": "95% CI ≈ [75.53, 83.73].", "meta": {"id": "WUA_STAT_25K_016128", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.27, μ=2.12, σ=3.26, compute z."}]}, "response": "z = 24.279.", "meta": {"id": "WUA_STAT_25K_016129", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.55, μ=43.87, σ=11.26, compute z."}]}, "response": "z = -2.160.", "meta": {"id": "WUA_STAT_25K_016130", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.48, compute E and Var."}]}, "response": "E[X]=5.76; Var[X]=2.995.", "meta": {"id": "WUA_STAT_25K_016131", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.98, s=38.29, n=39, compute CI."}]}, "response": "95% CI ≈ [145.96, 170.00].", "meta": {"id": "WUA_STAT_25K_016132", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.209, sens=0.89, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.797.", "meta": {"id": "WUA_STAT_25K_016133", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.12, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=2.218.", "meta": {"id": "WUA_STAT_25K_016134", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.19, compute E and Var."}]}, "response": "E[X]=2.85; Var[X]=2.309.", "meta": {"id": "WUA_STAT_25K_016135", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.64, s=3.17, n=162, compute CI."}]}, "response": "95% CI ≈ [35.15, 36.13].", "meta": {"id": "WUA_STAT_25K_016136", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.88, μ=19.28, σ=7.69, compute z."}]}, "response": "z = -0.442.", "meta": {"id": "WUA_STAT_25K_016137", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.94, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.402.", "meta": {"id": "WUA_STAT_25K_016138", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.74, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.358.", "meta": {"id": "WUA_STAT_25K_016139", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.38, s=2.37, n=118, compute CI."}]}, "response": "95% CI ≈ [149.95, 150.81].", "meta": {"id": "WUA_STAT_25K_016140", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.79, s=18.84, n=90, compute CI."}]}, "response": "95% CI ≈ [141.90, 149.68].", "meta": {"id": "WUA_STAT_25K_016141", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.24, μ=51.0, σ=18.5, compute z."}]}, "response": "z = -0.690.", "meta": {"id": "WUA_STAT_25K_016142", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.03, μ=58.78, σ=20.92, compute z."}]}, "response": "z = -1.948.", "meta": {"id": "WUA_STAT_25K_016143", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.39, compute E and Var."}]}, "response": "E[X]=10.53; Var[X]=6.423.", "meta": {"id": "WUA_STAT_25K_016144", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=47.03, s=9.95, n=138, compute CI."}]}, "response": "95% CI ≈ [45.37, 48.69].", "meta": {"id": "WUA_STAT_25K_016145", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.27, compute E and Var."}]}, "response": "E[X]=3.78; Var[X]=2.759.", "meta": {"id": "WUA_STAT_25K_016146", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.67, compute E and Var."}]}, "response": "E[X]=14.07; Var[X]=4.643.", "meta": {"id": "WUA_STAT_25K_016147", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.41, s=22.03, n=77, compute CI."}]}, "response": "95% CI ≈ [140.49, 150.33].", "meta": {"id": "WUA_STAT_25K_016148", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.56, compute E and Var."}]}, "response": "E[X]=20.72; Var[X]=9.117.", "meta": {"id": "WUA_STAT_25K_016149", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.03, μ=64.82, σ=21.08, compute z."}]}, "response": "z = 0.152.", "meta": {"id": "WUA_STAT_25K_016150", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.233, sens=0.97, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.551.", "meta": {"id": "WUA_STAT_25K_016151", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.79, s=16.46, n=43, compute CI."}]}, "response": "95% CI ≈ [183.87, 193.71].", "meta": {"id": "WUA_STAT_25K_016152", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.19, compute E and Var."}]}, "response": "E[X]=2.85; Var[X]=2.309.", "meta": {"id": "WUA_STAT_25K_016153", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.124, sens=0.91, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.539.", "meta": {"id": "WUA_STAT_25K_016154", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.57, compute E and Var."}]}, "response": "E[X]=5.70; Var[X]=2.451.", "meta": {"id": "WUA_STAT_25K_016155", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=99.47, μ=22.86, σ=8.37, compute z."}]}, "response": "z = 9.153.", "meta": {"id": "WUA_STAT_25K_016156", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=149.06, s=31.21, n=186, compute CI."}]}, "response": "95% CI ≈ [144.57, 153.55].", "meta": {"id": "WUA_STAT_25K_016157", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.12, compute E and Var."}]}, "response": "E[X]=4.44; Var[X]=3.907.", "meta": {"id": "WUA_STAT_25K_016158", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.79, compute E and Var."}]}, "response": "E[X]=20.54; Var[X]=4.313.", "meta": {"id": "WUA_STAT_25K_016159", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.92, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.475.", "meta": {"id": "WUA_STAT_25K_016160", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=24.51, s=34.67, n=28, compute CI."}]}, "response": "95% CI ≈ [11.67, 37.35].", "meta": {"id": "WUA_STAT_25K_016161", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.85, μ=22.46, σ=1.09, compute z."}]}, "response": "z = 7.697.", "meta": {"id": "WUA_STAT_25K_016162", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.78, μ=21.07, σ=15.94, compute z."}]}, "response": "z = 2.303.", "meta": {"id": "WUA_STAT_25K_016163", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.1, compute E and Var."}]}, "response": "E[X]=1.40; Var[X]=1.260.", "meta": {"id": "WUA_STAT_25K_016164", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=54.27, μ=38.19, σ=15.19, compute z."}]}, "response": "z = 1.059.", "meta": {"id": "WUA_STAT_25K_016165", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.74, compute E and Var."}]}, "response": "E[X]=12.58; Var[X]=3.271.", "meta": {"id": "WUA_STAT_25K_016166", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.43, compute E and Var."}]}, "response": "E[X]=9.89; Var[X]=5.637.", "meta": {"id": "WUA_STAT_25K_016167", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.038, sens=0.72, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.106.", "meta": {"id": "WUA_STAT_25K_016168", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.27, s=13.21, n=114, compute CI."}]}, "response": "95% CI ≈ [10.85, 15.69].", "meta": {"id": "WUA_STAT_25K_016169", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.48, s=31.48, n=122, compute CI."}]}, "response": "95% CI ≈ [71.89, 83.07].", "meta": {"id": "WUA_STAT_25K_016170", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.33, compute E and Var."}]}, "response": "E[X]=1.65; Var[X]=1.105.", "meta": {"id": "WUA_STAT_25K_016171", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.33, compute E and Var."}]}, "response": "E[X]=8.58; Var[X]=5.749.", "meta": {"id": "WUA_STAT_25K_016172", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.174, sens=0.79, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.510.", "meta": {"id": "WUA_STAT_25K_016173", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=115.94, s=7.07, n=63, compute CI."}]}, "response": "95% CI ≈ [114.19, 117.69].", "meta": {"id": "WUA_STAT_25K_016174", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.36, compute E and Var."}]}, "response": "E[X]=3.24; Var[X]=2.074.", "meta": {"id": "WUA_STAT_25K_016175", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.05, sens=0.96, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.153.", "meta": {"id": "WUA_STAT_25K_016176", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.6, s=29.41, n=167, compute CI."}]}, "response": "95% CI ≈ [70.14, 79.06].", "meta": {"id": "WUA_STAT_25K_016177", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.38, compute E and Var."}]}, "response": "E[X]=3.80; Var[X]=2.356.", "meta": {"id": "WUA_STAT_25K_016178", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.063, sens=0.87, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.226.", "meta": {"id": "WUA_STAT_25K_016179", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.246, sens=0.78, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.476.", "meta": {"id": "WUA_STAT_25K_016180", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=71.23, μ=22.85, σ=24.59, compute z."}]}, "response": "z = 1.967.", "meta": {"id": "WUA_STAT_25K_016181", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.076, sens=0.88, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.508.", "meta": {"id": "WUA_STAT_25K_016182", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.62, s=34.33, n=109, compute CI."}]}, "response": "95% CI ≈ [80.18, 93.06].", "meta": {"id": "WUA_STAT_25K_016183", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.7, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.241.", "meta": {"id": "WUA_STAT_25K_016184", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.57, s=4.18, n=24, compute CI."}]}, "response": "95% CI ≈ [72.90, 76.24].", "meta": {"id": "WUA_STAT_25K_016185", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=125.48, s=31.69, n=10, compute CI."}]}, "response": "95% CI ≈ [105.84, 145.12].", "meta": {"id": "WUA_STAT_25K_016186", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.71, compute E and Var."}]}, "response": "E[X]=4.26; Var[X]=1.235.", "meta": {"id": "WUA_STAT_25K_016187", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.84, s=29.2, n=52, compute CI."}]}, "response": "95% CI ≈ [174.90, 190.78].", "meta": {"id": "WUA_STAT_25K_016188", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.4, s=39.97, n=40, compute CI."}]}, "response": "95% CI ≈ [24.01, 48.79].", "meta": {"id": "WUA_STAT_25K_016189", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.011, sens=0.87, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.039.", "meta": {"id": "WUA_STAT_25K_016190", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.59, s=2.26, n=193, compute CI."}]}, "response": "95% CI ≈ [75.27, 75.91].", "meta": {"id": "WUA_STAT_25K_016191", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.43, s=12.57, n=105, compute CI."}]}, "response": "95% CI ≈ [58.03, 62.83].", "meta": {"id": "WUA_STAT_25K_016192", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.61, s=6.02, n=86, compute CI."}]}, "response": "95% CI ≈ [141.34, 143.88].", "meta": {"id": "WUA_STAT_25K_016193", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.131, sens=0.72, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.475.", "meta": {"id": "WUA_STAT_25K_016194", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.06, s=14.28, n=152, compute CI."}]}, "response": "95% CI ≈ [52.79, 57.33].", "meta": {"id": "WUA_STAT_25K_016195", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.9, μ=63.59, σ=11.24, compute z."}]}, "response": "z = -3.086.", "meta": {"id": "WUA_STAT_25K_016196", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.47, μ=29.65, σ=4.98, compute z."}]}, "response": "z = 1.169.", "meta": {"id": "WUA_STAT_25K_016197", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.212, sens=0.88, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.467.", "meta": {"id": "WUA_STAT_25K_016198", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.06, compute E and Var."}]}, "response": "E[X]=1.62; Var[X]=1.523.", "meta": {"id": "WUA_STAT_25K_016199", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.87, s=24.65, n=18, compute CI."}]}, "response": "95% CI ≈ [24.48, 47.26].", "meta": {"id": "WUA_STAT_25K_016200", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.55, compute E and Var."}]}, "response": "E[X]=17.60; Var[X]=7.920.", "meta": {"id": "WUA_STAT_25K_016201", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.044, sens=0.93, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.192.", "meta": {"id": "WUA_STAT_25K_016202", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.24, μ=54.56, σ=13.67, compute z."}]}, "response": "z = -2.876.", "meta": {"id": "WUA_STAT_25K_016203", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.16, compute E and Var."}]}, "response": "E[X]=5.76; Var[X]=4.838.", "meta": {"id": "WUA_STAT_25K_016204", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=52.07, s=8.29, n=197, compute CI."}]}, "response": "95% CI ≈ [50.91, 53.23].", "meta": {"id": "WUA_STAT_25K_016205", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=16.9, s=2.87, n=10, compute CI."}]}, "response": "95% CI ≈ [15.12, 18.68].", "meta": {"id": "WUA_STAT_25K_016206", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.83, s=21.23, n=150, compute CI."}]}, "response": "95% CI ≈ [50.43, 57.23].", "meta": {"id": "WUA_STAT_25K_016207", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.31, s=39.3, n=85, compute CI."}]}, "response": "95% CI ≈ [167.96, 184.66].", "meta": {"id": "WUA_STAT_25K_016208", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.28, μ=27.73, σ=9.05, compute z."}]}, "response": "z = 2.492.", "meta": {"id": "WUA_STAT_25K_016209", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.71, compute E and Var."}]}, "response": "E[X]=12.07; Var[X]=3.500.", "meta": {"id": "WUA_STAT_25K_016210", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.16, μ=15.44, σ=5.87, compute z."}]}, "response": "z = 4.041.", "meta": {"id": "WUA_STAT_25K_016211", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.73, μ=1.32, σ=11.11, compute z."}]}, "response": "z = 6.518.", "meta": {"id": "WUA_STAT_25K_016212", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.8, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.472.", "meta": {"id": "WUA_STAT_25K_016213", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=140.02, s=38.45, n=101, compute CI."}]}, "response": "95% CI ≈ [132.52, 147.52].", "meta": {"id": "WUA_STAT_25K_016214", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.17, compute E and Var."}]}, "response": "E[X]=3.74; Var[X]=3.104.", "meta": {"id": "WUA_STAT_25K_016215", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=36.51, s=23.25, n=101, compute CI."}]}, "response": "95% CI ≈ [31.98, 41.04].", "meta": {"id": "WUA_STAT_25K_016216", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.83, s=37.79, n=72, compute CI."}]}, "response": "95% CI ≈ [54.10, 71.56].", "meta": {"id": "WUA_STAT_25K_016217", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.172, sens=0.76, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.798.", "meta": {"id": "WUA_STAT_25K_016218", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.31, μ=32.36, σ=10.31, compute z."}]}, "response": "z = 2.808.", "meta": {"id": "WUA_STAT_25K_016219", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.066, sens=0.85, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.667.", "meta": {"id": "WUA_STAT_25K_016220", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.096, sens=0.71, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.790.", "meta": {"id": "WUA_STAT_25K_016221", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.22, μ=92.35, σ=18.19, compute z."}]}, "response": "z = -0.612.", "meta": {"id": "WUA_STAT_25K_016222", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.55, μ=37.97, σ=10.89, compute z."}]}, "response": "z = -0.314.", "meta": {"id": "WUA_STAT_25K_016223", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.7, compute E and Var."}]}, "response": "E[X]=25.90; Var[X]=7.770.", "meta": {"id": "WUA_STAT_25K_016224", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.04, s=28.0, n=55, compute CI."}]}, "response": "95% CI ≈ [54.64, 69.44].", "meta": {"id": "WUA_STAT_25K_016225", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=85.72, s=9.19, n=37, compute CI."}]}, "response": "95% CI ≈ [82.76, 88.68].", "meta": {"id": "WUA_STAT_25K_016226", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.78, compute E and Var."}]}, "response": "E[X]=30.42; Var[X]=6.692.", "meta": {"id": "WUA_STAT_25K_016227", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.088, sens=0.82, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.798.", "meta": {"id": "WUA_STAT_25K_016228", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.29, s=7.71, n=115, compute CI."}]}, "response": "95% CI ≈ [16.88, 19.70].", "meta": {"id": "WUA_STAT_25K_016229", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.42, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=4.872.", "meta": {"id": "WUA_STAT_25K_016230", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=26.84, s=39.53, n=94, compute CI."}]}, "response": "95% CI ≈ [18.85, 34.83].", "meta": {"id": "WUA_STAT_25K_016231", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.3, μ=14.82, σ=21.5, compute z."}]}, "response": "z = 0.906.", "meta": {"id": "WUA_STAT_25K_016232", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=30.53, μ=58.36, σ=15.16, compute z."}]}, "response": "z = -1.836.", "meta": {"id": "WUA_STAT_25K_016233", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=5.99, μ=99.63, σ=23.03, compute z."}]}, "response": "z = -4.066.", "meta": {"id": "WUA_STAT_25K_016234", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.94, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.892.", "meta": {"id": "WUA_STAT_25K_016235", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=94.84, μ=99.27, σ=17.86, compute z."}]}, "response": "z = -0.248.", "meta": {"id": "WUA_STAT_25K_016236", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.31, μ=99.07, σ=23.56, compute z."}]}, "response": "z = -2.706.", "meta": {"id": "WUA_STAT_25K_016237", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.09, μ=43.68, σ=24.08, compute z."}]}, "response": "z = 1.969.", "meta": {"id": "WUA_STAT_25K_016238", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.84, s=2.7, n=42, compute CI."}]}, "response": "95% CI ≈ [156.02, 157.66].", "meta": {"id": "WUA_STAT_25K_016239", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.98, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.519.", "meta": {"id": "WUA_STAT_25K_016240", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.22, μ=60.56, σ=9.23, compute z."}]}, "response": "z = -2.854.", "meta": {"id": "WUA_STAT_25K_016241", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.73, μ=52.55, σ=7.51, compute z."}]}, "response": "z = 4.818.", "meta": {"id": "WUA_STAT_25K_016242", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.18, s=32.33, n=135, compute CI."}]}, "response": "95% CI ≈ [7.73, 18.63].", "meta": {"id": "WUA_STAT_25K_016243", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.69, compute E and Var."}]}, "response": "E[X]=11.73; Var[X]=3.636.", "meta": {"id": "WUA_STAT_25K_016244", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.249, sens=0.92, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.604.", "meta": {"id": "WUA_STAT_25K_016245", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.83, s=39.09, n=85, compute CI."}]}, "response": "95% CI ≈ [102.52, 119.14].", "meta": {"id": "WUA_STAT_25K_016246", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.112, sens=0.73, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.351.", "meta": {"id": "WUA_STAT_25K_016247", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.161, sens=0.93, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.665.", "meta": {"id": "WUA_STAT_25K_016248", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.077, sens=0.88, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.400.", "meta": {"id": "WUA_STAT_25K_016249", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.26, compute E and Var."}]}, "response": "E[X]=3.64; Var[X]=2.694.", "meta": {"id": "WUA_STAT_25K_016250", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.77, compute E and Var."}]}, "response": "E[X]=20.02; Var[X]=4.605.", "meta": {"id": "WUA_STAT_25K_016251", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.1, μ=39.87, σ=15.48, compute z."}]}, "response": "z = -0.179.", "meta": {"id": "WUA_STAT_25K_016252", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.29, compute E and Var."}]}, "response": "E[X]=2.61; Var[X]=1.853.", "meta": {"id": "WUA_STAT_25K_016253", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=97.75, s=10.24, n=144, compute CI."}]}, "response": "95% CI ≈ [96.08, 99.42].", "meta": {"id": "WUA_STAT_25K_016254", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.105, sens=0.72, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.226.", "meta": {"id": "WUA_STAT_25K_016255", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.38, compute E and Var."}]}, "response": "E[X]=14.82; Var[X]=9.188.", "meta": {"id": "WUA_STAT_25K_016256", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=82.95, s=2.6, n=131, compute CI."}]}, "response": "95% CI ≈ [82.50, 83.40].", "meta": {"id": "WUA_STAT_25K_016257", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.111, sens=0.74, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.339.", "meta": {"id": "WUA_STAT_25K_016258", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.65, compute E and Var."}]}, "response": "E[X]=9.75; Var[X]=3.412.", "meta": {"id": "WUA_STAT_25K_016259", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.83, μ=29.15, σ=16.25, compute z."}]}, "response": "z = 1.826.", "meta": {"id": "WUA_STAT_25K_016260", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.19, compute E and Var."}]}, "response": "E[X]=4.75; Var[X]=3.848.", "meta": {"id": "WUA_STAT_25K_016261", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.86, s=28.18, n=145, compute CI."}]}, "response": "95% CI ≈ [178.27, 187.45].", "meta": {"id": "WUA_STAT_25K_016262", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=96.03, s=15.49, n=87, compute CI."}]}, "response": "95% CI ≈ [92.78, 99.28].", "meta": {"id": "WUA_STAT_25K_016263", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=128.09, s=34.23, n=74, compute CI."}]}, "response": "95% CI ≈ [120.29, 135.89].", "meta": {"id": "WUA_STAT_25K_016264", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.064, sens=0.75, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.243.", "meta": {"id": "WUA_STAT_25K_016265", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.38, μ=67.24, σ=15.55, compute z."}]}, "response": "z = -0.377.", "meta": {"id": "WUA_STAT_25K_016266", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.33, compute E and Var."}]}, "response": "E[X]=6.93; Var[X]=4.643.", "meta": {"id": "WUA_STAT_25K_016267", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=4.12, μ=30.34, σ=15.01, compute z."}]}, "response": "z = -1.747.", "meta": {"id": "WUA_STAT_25K_016268", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.33, μ=69.93, σ=14.6, compute z."}]}, "response": "z = 0.301.", "meta": {"id": "WUA_STAT_25K_016269", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.85, s=31.48, n=124, compute CI."}]}, "response": "95% CI ≈ [48.31, 59.39].", "meta": {"id": "WUA_STAT_25K_016270", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.98, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.601.", "meta": {"id": "WUA_STAT_25K_016271", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=105.39, s=9.59, n=72, compute CI."}]}, "response": "95% CI ≈ [103.17, 107.61].", "meta": {"id": "WUA_STAT_25K_016272", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=26.82, s=18.39, n=31, compute CI."}]}, "response": "95% CI ≈ [20.35, 33.29].", "meta": {"id": "WUA_STAT_25K_016273", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.2, sens=0.95, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.450.", "meta": {"id": "WUA_STAT_25K_016274", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=176.22, s=15.63, n=15, compute CI."}]}, "response": "95% CI ≈ [168.31, 184.13].", "meta": {"id": "WUA_STAT_25K_016275", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.38, compute E and Var."}]}, "response": "E[X]=3.42; Var[X]=2.120.", "meta": {"id": "WUA_STAT_25K_016276", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=66.32, μ=2.2, σ=12.94, compute z."}]}, "response": "z = 4.955.", "meta": {"id": "WUA_STAT_25K_016277", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.67, compute E and Var."}]}, "response": "E[X]=4.02; Var[X]=1.327.", "meta": {"id": "WUA_STAT_25K_016278", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.01, s=9.94, n=54, compute CI."}]}, "response": "95% CI ≈ [191.36, 196.66].", "meta": {"id": "WUA_STAT_25K_016279", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.73, μ=8.99, σ=21.54, compute z."}]}, "response": "z = 3.006.", "meta": {"id": "WUA_STAT_25K_016280", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.88, μ=4.69, σ=12.14, compute z."}]}, "response": "z = 2.487.", "meta": {"id": "WUA_STAT_25K_016281", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.059, sens=0.71, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.218.", "meta": {"id": "WUA_STAT_25K_016282", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.41, compute E and Var."}]}, "response": "E[X]=12.30; Var[X]=7.257.", "meta": {"id": "WUA_STAT_25K_016283", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.35, μ=12.56, σ=11.95, compute z."}]}, "response": "z = 3.413.", "meta": {"id": "WUA_STAT_25K_016284", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.37, s=2.22, n=54, compute CI."}]}, "response": "95% CI ≈ [134.78, 135.96].", "meta": {"id": "WUA_STAT_25K_016285", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=13, p=0.72, compute E and Var."}]}, "response": "E[X]=9.36; Var[X]=2.621.", "meta": {"id": "WUA_STAT_25K_016286", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.094, sens=0.84, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.744.", "meta": {"id": "WUA_STAT_25K_016287", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.241, sens=0.71, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.672.", "meta": {"id": "WUA_STAT_25K_016288", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.73, compute E and Var."}]}, "response": "E[X]=24.82; Var[X]=6.701.", "meta": {"id": "WUA_STAT_25K_016289", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.69, μ=99.46, σ=15.11, compute z."}]}, "response": "z = -0.051.", "meta": {"id": "WUA_STAT_25K_016290", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.29, s=9.4, n=159, compute CI."}]}, "response": "95% CI ≈ [180.83, 183.75].", "meta": {"id": "WUA_STAT_25K_016291", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.013, sens=0.95, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.048.", "meta": {"id": "WUA_STAT_25K_016292", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.59, compute E and Var."}]}, "response": "E[X]=12.39; Var[X]=5.080.", "meta": {"id": "WUA_STAT_25K_016293", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.77, compute E and Var."}]}, "response": "E[X]=18.48; Var[X]=4.250.", "meta": {"id": "WUA_STAT_25K_016294", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=25.85, μ=80.83, σ=12.97, compute z."}]}, "response": "z = -4.239.", "meta": {"id": "WUA_STAT_25K_016295", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.2, compute E and Var."}]}, "response": "E[X]=1.20; Var[X]=0.960.", "meta": {"id": "WUA_STAT_25K_016296", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.4, μ=7.87, σ=10.52, compute z."}]}, "response": "z = 4.613.", "meta": {"id": "WUA_STAT_25K_016297", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.35, s=4.09, n=19, compute CI."}]}, "response": "95% CI ≈ [78.51, 82.19].", "meta": {"id": "WUA_STAT_25K_016298", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.51, compute E and Var."}]}, "response": "E[X]=12.75; Var[X]=6.247.", "meta": {"id": "WUA_STAT_25K_016299", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.73, μ=25.2, σ=6.48, compute z."}]}, "response": "z = 10.267.", "meta": {"id": "WUA_STAT_25K_016300", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.98, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.470.", "meta": {"id": "WUA_STAT_25K_016301", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=13.6, s=26.2, n=88, compute CI."}]}, "response": "95% CI ≈ [8.13, 19.07].", "meta": {"id": "WUA_STAT_25K_016302", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.157, sens=0.88, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.396.", "meta": {"id": "WUA_STAT_25K_016303", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=47.68, μ=37.83, σ=14.75, compute z."}]}, "response": "z = 0.668.", "meta": {"id": "WUA_STAT_25K_016304", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.08, μ=97.54, σ=7.99, compute z."}]}, "response": "z = -11.071.", "meta": {"id": "WUA_STAT_25K_016305", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.9, s=1.13, n=89, compute CI."}]}, "response": "95% CI ≈ [43.67, 44.13].", "meta": {"id": "WUA_STAT_25K_016306", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.55, s=10.09, n=34, compute CI."}]}, "response": "95% CI ≈ [47.16, 53.94].", "meta": {"id": "WUA_STAT_25K_016307", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=60.26, s=14.81, n=90, compute CI."}]}, "response": "95% CI ≈ [57.20, 63.32].", "meta": {"id": "WUA_STAT_25K_016308", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.38, μ=55.25, σ=8.79, compute z."}]}, "response": "z = -3.967.", "meta": {"id": "WUA_STAT_25K_016309", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.23, s=13.71, n=182, compute CI."}]}, "response": "95% CI ≈ [178.24, 182.22].", "meta": {"id": "WUA_STAT_25K_016310", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=78.92, s=1.25, n=172, compute CI."}]}, "response": "95% CI ≈ [78.73, 79.11].", "meta": {"id": "WUA_STAT_25K_016311", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.1, compute E and Var."}]}, "response": "E[X]=3.60; Var[X]=3.240.", "meta": {"id": "WUA_STAT_25K_016312", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.71, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.339.", "meta": {"id": "WUA_STAT_25K_016313", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.44, compute E and Var."}]}, "response": "E[X]=14.96; Var[X]=8.378.", "meta": {"id": "WUA_STAT_25K_016314", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.39, μ=31.54, σ=10.96, compute z."}]}, "response": "z = 3.910.", "meta": {"id": "WUA_STAT_25K_016315", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.46, s=32.24, n=118, compute CI."}]}, "response": "95% CI ≈ [77.64, 89.28].", "meta": {"id": "WUA_STAT_25K_016316", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=121.02, s=14.03, n=25, compute CI."}]}, "response": "95% CI ≈ [115.52, 126.52].", "meta": {"id": "WUA_STAT_25K_016317", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.21, sens=0.83, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.580.", "meta": {"id": "WUA_STAT_25K_016318", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.75, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.423.", "meta": {"id": "WUA_STAT_25K_016319", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.37, compute E and Var."}]}, "response": "E[X]=7.77; Var[X]=4.895.", "meta": {"id": "WUA_STAT_25K_016320", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=126.1, s=7.9, n=96, compute CI."}]}, "response": "95% CI ≈ [124.52, 127.68].", "meta": {"id": "WUA_STAT_25K_016321", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.89, μ=98.37, σ=4.28, compute z."}]}, "response": "z = -3.150.", "meta": {"id": "WUA_STAT_25K_016322", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.153, sens=0.94, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.586.", "meta": {"id": "WUA_STAT_25K_016323", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.07, sens=0.71, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.195.", "meta": {"id": "WUA_STAT_25K_016324", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.233, sens=0.87, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.477.", "meta": {"id": "WUA_STAT_25K_016325", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.16, compute E and Var."}]}, "response": "E[X]=5.28; Var[X]=4.435.", "meta": {"id": "WUA_STAT_25K_016326", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.3, compute E and Var."}]}, "response": "E[X]=12.00; Var[X]=8.400.", "meta": {"id": "WUA_STAT_25K_016327", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.7, compute E and Var."}]}, "response": "E[X]=21.70; Var[X]=6.510.", "meta": {"id": "WUA_STAT_25K_016328", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.65, compute E and Var."}]}, "response": "E[X]=3.90; Var[X]=1.365.", "meta": {"id": "WUA_STAT_25K_016329", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.8, μ=0.72, σ=21.46, compute z."}]}, "response": "z = 2.939.", "meta": {"id": "WUA_STAT_25K_016330", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.09, compute E and Var."}]}, "response": "E[X]=0.45; Var[X]=0.409.", "meta": {"id": "WUA_STAT_25K_016331", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.28, compute E and Var."}]}, "response": "E[X]=4.48; Var[X]=3.226.", "meta": {"id": "WUA_STAT_25K_016332", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.04, μ=14.51, σ=4.19, compute z."}]}, "response": "z = 0.126.", "meta": {"id": "WUA_STAT_25K_016333", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.34, s=18.24, n=142, compute CI."}]}, "response": "95% CI ≈ [56.34, 62.34].", "meta": {"id": "WUA_STAT_25K_016334", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.11, compute E and Var."}]}, "response": "E[X]=1.32; Var[X]=1.175.", "meta": {"id": "WUA_STAT_25K_016335", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.97, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.389.", "meta": {"id": "WUA_STAT_25K_016336", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.017, sens=0.87, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.059.", "meta": {"id": "WUA_STAT_25K_016337", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.25, compute E and Var."}]}, "response": "E[X]=2.75; Var[X]=2.062.", "meta": {"id": "WUA_STAT_25K_016338", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.06, s=21.01, n=44, compute CI."}]}, "response": "95% CI ≈ [150.85, 163.27].", "meta": {"id": "WUA_STAT_25K_016339", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=57.28, μ=53.02, σ=10.24, compute z."}]}, "response": "z = 0.416.", "meta": {"id": "WUA_STAT_25K_016340", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.73, μ=77.72, σ=6.81, compute z."}]}, "response": "z = 1.323.", "meta": {"id": "WUA_STAT_25K_016341", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.75, μ=97.69, σ=7.01, compute z."}]}, "response": "z = -7.695.", "meta": {"id": "WUA_STAT_25K_016342", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.77, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.434.", "meta": {"id": "WUA_STAT_25K_016343", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.12, compute E and Var."}]}, "response": "E[X]=2.40; Var[X]=2.112.", "meta": {"id": "WUA_STAT_25K_016344", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.7, compute E and Var."}]}, "response": "E[X]=16.10; Var[X]=4.830.", "meta": {"id": "WUA_STAT_25K_016345", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.95, μ=20.25, σ=9.86, compute z."}]}, "response": "z = 1.491.", "meta": {"id": "WUA_STAT_25K_016346", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.54, compute E and Var."}]}, "response": "E[X]=19.44; Var[X]=8.942.", "meta": {"id": "WUA_STAT_25K_016347", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.13, s=9.18, n=200, compute CI."}]}, "response": "95% CI ≈ [57.86, 60.40].", "meta": {"id": "WUA_STAT_25K_016348", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.5, μ=93.88, σ=13.76, compute z."}]}, "response": "z = -3.225.", "meta": {"id": "WUA_STAT_25K_016349", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.021, sens=0.72, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.052.", "meta": {"id": "WUA_STAT_25K_016350", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.12, s=10.08, n=95, compute CI."}]}, "response": "95% CI ≈ [90.09, 94.15].", "meta": {"id": "WUA_STAT_25K_016351", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.75, μ=21.26, σ=15.77, compute z."}]}, "response": "z = 1.743.", "meta": {"id": "WUA_STAT_25K_016352", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.16, compute E and Var."}]}, "response": "E[X]=5.92; Var[X]=4.973.", "meta": {"id": "WUA_STAT_25K_016353", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.53, compute E and Var."}]}, "response": "E[X]=5.83; Var[X]=2.740.", "meta": {"id": "WUA_STAT_25K_016354", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=91.83, μ=37.18, σ=12.5, compute z."}]}, "response": "z = 4.372.", "meta": {"id": "WUA_STAT_25K_016355", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=145.49, s=19.74, n=186, compute CI."}]}, "response": "95% CI ≈ [142.65, 148.33].", "meta": {"id": "WUA_STAT_25K_016356", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.17, compute E and Var."}]}, "response": "E[X]=4.25; Var[X]=3.527.", "meta": {"id": "WUA_STAT_25K_016357", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.063, sens=0.87, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.184.", "meta": {"id": "WUA_STAT_25K_016358", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.17, μ=1.52, σ=15.01, compute z."}]}, "response": "z = 5.373.", "meta": {"id": "WUA_STAT_25K_016359", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.53, compute E and Var."}]}, "response": "E[X]=13.78; Var[X]=6.477.", "meta": {"id": "WUA_STAT_25K_016360", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.108, sens=0.94, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.387.", "meta": {"id": "WUA_STAT_25K_016361", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.15, compute E and Var."}]}, "response": "E[X]=5.10; Var[X]=4.335.", "meta": {"id": "WUA_STAT_25K_016362", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=74.93, μ=48.99, σ=4.63, compute z."}]}, "response": "z = 5.603.", "meta": {"id": "WUA_STAT_25K_016363", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.22, μ=69.47, σ=9.9, compute z."}]}, "response": "z = -4.672.", "meta": {"id": "WUA_STAT_25K_016364", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.39, compute E and Var."}]}, "response": "E[X]=8.19; Var[X]=4.996.", "meta": {"id": "WUA_STAT_25K_016365", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.31, μ=97.68, σ=19.14, compute z."}]}, "response": "z = -2.893.", "meta": {"id": "WUA_STAT_25K_016366", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.99, μ=55.95, σ=8.41, compute z."}]}, "response": "z = 4.999.", "meta": {"id": "WUA_STAT_25K_016367", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.134, sens=0.84, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.325.", "meta": {"id": "WUA_STAT_25K_016368", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.71, compute E and Var."}]}, "response": "E[X]=14.91; Var[X]=4.324.", "meta": {"id": "WUA_STAT_25K_016369", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.04, μ=59.7, σ=1.19, compute z."}]}, "response": "z = -26.605.", "meta": {"id": "WUA_STAT_25K_016370", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.98, s=5.94, n=171, compute CI."}]}, "response": "95% CI ≈ [135.09, 136.87].", "meta": {"id": "WUA_STAT_25K_016371", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.51, μ=44.56, σ=9.92, compute z."}]}, "response": "z = 1.205.", "meta": {"id": "WUA_STAT_25K_016372", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.213, sens=0.77, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.465.", "meta": {"id": "WUA_STAT_25K_016373", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.44, compute E and Var."}]}, "response": "E[X]=10.12; Var[X]=5.667.", "meta": {"id": "WUA_STAT_25K_016374", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.88, μ=56.49, σ=11.18, compute z."}]}, "response": "z = -1.486.", "meta": {"id": "WUA_STAT_25K_016375", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.07, compute E and Var."}]}, "response": "E[X]=2.45; Var[X]=2.279.", "meta": {"id": "WUA_STAT_25K_016376", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.58, s=38.96, n=180, compute CI."}]}, "response": "95% CI ≈ [43.89, 55.27].", "meta": {"id": "WUA_STAT_25K_016377", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.105, sens=0.71, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.481.", "meta": {"id": "WUA_STAT_25K_016378", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.77, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.367.", "meta": {"id": "WUA_STAT_25K_016379", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.73, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.361.", "meta": {"id": "WUA_STAT_25K_016380", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.35, μ=36.75, σ=12.82, compute z."}]}, "response": "z = 0.359.", "meta": {"id": "WUA_STAT_25K_016381", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.36, compute E and Var."}]}, "response": "E[X]=4.32; Var[X]=2.765.", "meta": {"id": "WUA_STAT_25K_016382", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=66.31, s=16.72, n=188, compute CI."}]}, "response": "95% CI ≈ [63.92, 68.70].", "meta": {"id": "WUA_STAT_25K_016383", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.14, compute E and Var."}]}, "response": "E[X]=4.62; Var[X]=3.973.", "meta": {"id": "WUA_STAT_25K_016384", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.248, sens=0.75, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.622.", "meta": {"id": "WUA_STAT_25K_016385", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.82, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.492.", "meta": {"id": "WUA_STAT_25K_016386", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=55.28, s=31.66, n=147, compute CI."}]}, "response": "95% CI ≈ [50.16, 60.40].", "meta": {"id": "WUA_STAT_25K_016387", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.07, μ=88.08, σ=3.7, compute z."}]}, "response": "z = -23.516.", "meta": {"id": "WUA_STAT_25K_016388", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=167.99, s=39.58, n=15, compute CI."}]}, "response": "95% CI ≈ [147.96, 188.02].", "meta": {"id": "WUA_STAT_25K_016389", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.39, compute E and Var."}]}, "response": "E[X]=8.58; Var[X]=5.234.", "meta": {"id": "WUA_STAT_25K_016390", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=157.18, s=27.45, n=32, compute CI."}]}, "response": "95% CI ≈ [147.67, 166.69].", "meta": {"id": "WUA_STAT_25K_016391", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.18, μ=73.64, σ=17.05, compute z."}]}, "response": "z = -3.605.", "meta": {"id": "WUA_STAT_25K_016392", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=26.83, μ=97.14, σ=15.06, compute z."}]}, "response": "z = -4.669.", "meta": {"id": "WUA_STAT_25K_016393", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.17, s=34.02, n=31, compute CI."}]}, "response": "95% CI ≈ [21.19, 45.15].", "meta": {"id": "WUA_STAT_25K_016394", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.217, sens=0.9, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.462.", "meta": {"id": "WUA_STAT_25K_016395", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=39.15, μ=19.92, σ=11.02, compute z."}]}, "response": "z = 1.745.", "meta": {"id": "WUA_STAT_25K_016396", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.187, sens=0.76, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.660.", "meta": {"id": "WUA_STAT_25K_016397", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.95, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.375.", "meta": {"id": "WUA_STAT_25K_016398", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=74.93, s=18.41, n=183, compute CI."}]}, "response": "95% CI ≈ [72.26, 77.60].", "meta": {"id": "WUA_STAT_25K_016399", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.51, μ=3.77, σ=1.89, compute z."}]}, "response": "z = 18.381.", "meta": {"id": "WUA_STAT_25K_016400", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.64, compute E and Var."}]}, "response": "E[X]=5.12; Var[X]=1.843.", "meta": {"id": "WUA_STAT_25K_016401", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.29, compute E and Var."}]}, "response": "E[X]=6.96; Var[X]=4.942.", "meta": {"id": "WUA_STAT_25K_016402", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.7, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.556.", "meta": {"id": "WUA_STAT_25K_016403", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.075, sens=0.98, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.665.", "meta": {"id": "WUA_STAT_25K_016404", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.117, sens=0.77, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.836.", "meta": {"id": "WUA_STAT_25K_016405", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.22, compute E and Var."}]}, "response": "E[X]=5.06; Var[X]=3.947.", "meta": {"id": "WUA_STAT_25K_016406", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.207, sens=0.71, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.861.", "meta": {"id": "WUA_STAT_25K_016407", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.52, compute E and Var."}]}, "response": "E[X]=18.20; Var[X]=8.736.", "meta": {"id": "WUA_STAT_25K_016408", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.04, μ=82.27, σ=8.55, compute z."}]}, "response": "z = -8.799.", "meta": {"id": "WUA_STAT_25K_016409", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.36, μ=43.89, σ=1.0, compute z."}]}, "response": "z = -14.530.", "meta": {"id": "WUA_STAT_25K_016410", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.091, sens=0.9, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.530.", "meta": {"id": "WUA_STAT_25K_016411", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.53, s=32.28, n=119, compute CI."}]}, "response": "95% CI ≈ [38.73, 50.33].", "meta": {"id": "WUA_STAT_25K_016412", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.07, μ=16.46, σ=16.82, compute z."}]}, "response": "z = -0.439.", "meta": {"id": "WUA_STAT_25K_016413", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.48, s=7.63, n=19, compute CI."}]}, "response": "95% CI ≈ [96.05, 102.91].", "meta": {"id": "WUA_STAT_25K_016414", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.012, sens=0.84, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.035.", "meta": {"id": "WUA_STAT_25K_016415", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.78, s=20.32, n=174, compute CI."}]}, "response": "95% CI ≈ [35.76, 41.80].", "meta": {"id": "WUA_STAT_25K_016416", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.12, compute E and Var."}]}, "response": "E[X]=1.68; Var[X]=1.478.", "meta": {"id": "WUA_STAT_25K_016417", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.49, s=9.12, n=122, compute CI."}]}, "response": "95% CI ≈ [129.87, 133.11].", "meta": {"id": "WUA_STAT_25K_016418", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.77, μ=17.52, σ=5.57, compute z."}]}, "response": "z = 3.636.", "meta": {"id": "WUA_STAT_25K_016419", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.25, compute E and Var."}]}, "response": "E[X]=3.50; Var[X]=2.625.", "meta": {"id": "WUA_STAT_25K_016420", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=14.21, μ=70.6, σ=1.28, compute z."}]}, "response": "z = -44.055.", "meta": {"id": "WUA_STAT_25K_016421", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.071, sens=0.84, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.204.", "meta": {"id": "WUA_STAT_25K_016422", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.08, sens=0.73, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.366.", "meta": {"id": "WUA_STAT_25K_016423", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.19, s=27.15, n=73, compute CI."}]}, "response": "95% CI ≈ [167.96, 180.42].", "meta": {"id": "WUA_STAT_25K_016424", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.101, sens=0.85, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.248.", "meta": {"id": "WUA_STAT_25K_016425", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.145, sens=0.77, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.310.", "meta": {"id": "WUA_STAT_25K_016426", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.74, compute E and Var."}]}, "response": "E[X]=25.16; Var[X]=6.542.", "meta": {"id": "WUA_STAT_25K_016427", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.029, sens=0.85, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.092.", "meta": {"id": "WUA_STAT_25K_016428", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.156, sens=0.73, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.551.", "meta": {"id": "WUA_STAT_25K_016429", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.17, compute E and Var."}]}, "response": "E[X]=6.29; Var[X]=5.221.", "meta": {"id": "WUA_STAT_25K_016430", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.087, sens=0.72, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.433.", "meta": {"id": "WUA_STAT_25K_016431", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.09, s=37.61, n=137, compute CI."}]}, "response": "95% CI ≈ [181.79, 194.39].", "meta": {"id": "WUA_STAT_25K_016432", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.68, compute E and Var."}]}, "response": "E[X]=27.20; Var[X]=8.704.", "meta": {"id": "WUA_STAT_25K_016433", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.6, μ=28.29, σ=4.98, compute z."}]}, "response": "z = -5.560.", "meta": {"id": "WUA_STAT_25K_016434", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=56.75, s=28.27, n=12, compute CI."}]}, "response": "95% CI ≈ [40.75, 72.75].", "meta": {"id": "WUA_STAT_25K_016435", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.25, s=29.22, n=133, compute CI."}]}, "response": "95% CI ≈ [122.28, 132.22].", "meta": {"id": "WUA_STAT_25K_016436", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.234, sens=0.95, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.527.", "meta": {"id": "WUA_STAT_25K_016437", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.62, μ=70.11, σ=7.66, compute z."}]}, "response": "z = -6.591.", "meta": {"id": "WUA_STAT_25K_016438", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.96, μ=73.87, σ=18.36, compute z."}]}, "response": "z = -1.466.", "meta": {"id": "WUA_STAT_25K_016439", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.96, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.659.", "meta": {"id": "WUA_STAT_25K_016440", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.55, compute E and Var."}]}, "response": "E[X]=12.65; Var[X]=5.692.", "meta": {"id": "WUA_STAT_25K_016441", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.62, compute E and Var."}]}, "response": "E[X]=14.26; Var[X]=5.419.", "meta": {"id": "WUA_STAT_25K_016442", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=117.94, s=3.3, n=107, compute CI."}]}, "response": "95% CI ≈ [117.31, 118.57].", "meta": {"id": "WUA_STAT_25K_016443", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.77, s=19.19, n=109, compute CI."}]}, "response": "95% CI ≈ [8.17, 15.37].", "meta": {"id": "WUA_STAT_25K_016444", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.77, s=18.55, n=17, compute CI."}]}, "response": "95% CI ≈ [89.95, 107.59].", "meta": {"id": "WUA_STAT_25K_016445", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.41, μ=8.94, σ=7.19, compute z."}]}, "response": "z = 9.801.", "meta": {"id": "WUA_STAT_25K_016446", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=59.95, s=26.96, n=128, compute CI."}]}, "response": "95% CI ≈ [55.28, 64.62].", "meta": {"id": "WUA_STAT_25K_016447", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.63, compute E and Var."}]}, "response": "E[X]=21.42; Var[X]=7.925.", "meta": {"id": "WUA_STAT_25K_016448", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.79, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.337.", "meta": {"id": "WUA_STAT_25K_016449", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=3.95, μ=52.28, σ=15.34, compute z."}]}, "response": "z = -3.151.", "meta": {"id": "WUA_STAT_25K_016450", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.22, s=22.4, n=33, compute CI."}]}, "response": "95% CI ≈ [99.58, 114.86].", "meta": {"id": "WUA_STAT_25K_016451", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.239, sens=0.73, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.884.", "meta": {"id": "WUA_STAT_25K_016452", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.81, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.477.", "meta": {"id": "WUA_STAT_25K_016453", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.35, compute E and Var."}]}, "response": "E[X]=7.35; Var[X]=4.777.", "meta": {"id": "WUA_STAT_25K_016454", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.023, sens=0.71, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.074.", "meta": {"id": "WUA_STAT_25K_016455", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.11, s=2.11, n=73, compute CI."}]}, "response": "95% CI ≈ [162.63, 163.59].", "meta": {"id": "WUA_STAT_25K_016456", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.41, compute E and Var."}]}, "response": "E[X]=3.28; Var[X]=1.935.", "meta": {"id": "WUA_STAT_25K_016457", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=49.08, s=7.04, n=145, compute CI."}]}, "response": "95% CI ≈ [47.93, 50.23].", "meta": {"id": "WUA_STAT_25K_016458", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.115, sens=0.85, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.298.", "meta": {"id": "WUA_STAT_25K_016459", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.222, sens=0.89, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.927.", "meta": {"id": "WUA_STAT_25K_016460", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=46.83, μ=89.95, σ=3.18, compute z."}]}, "response": "z = -13.560.", "meta": {"id": "WUA_STAT_25K_016461", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.95, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.434.", "meta": {"id": "WUA_STAT_25K_016462", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.65, compute E and Var."}]}, "response": "E[X]=24.70; Var[X]=8.645.", "meta": {"id": "WUA_STAT_25K_016463", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.26, μ=97.93, σ=4.49, compute z."}]}, "response": "z = -14.180.", "meta": {"id": "WUA_STAT_25K_016464", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=43.55, s=25.93, n=37, compute CI."}]}, "response": "95% CI ≈ [35.19, 51.91].", "meta": {"id": "WUA_STAT_25K_016465", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.14, μ=14.56, σ=6.92, compute z."}]}, "response": "z = 0.228.", "meta": {"id": "WUA_STAT_25K_016466", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.62, compute E and Var."}]}, "response": "E[X]=4.34; Var[X]=1.649.", "meta": {"id": "WUA_STAT_25K_016467", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.23, s=23.52, n=75, compute CI."}]}, "response": "95% CI ≈ [77.91, 88.55].", "meta": {"id": "WUA_STAT_25K_016468", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.48, compute E and Var."}]}, "response": "E[X]=14.40; Var[X]=7.488.", "meta": {"id": "WUA_STAT_25K_016469", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.168, sens=0.97, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.907.", "meta": {"id": "WUA_STAT_25K_016470", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.36, compute E and Var."}]}, "response": "E[X]=8.28; Var[X]=5.299.", "meta": {"id": "WUA_STAT_25K_016471", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=19.4, s=20.0, n=46, compute CI."}]}, "response": "95% CI ≈ [13.62, 25.18].", "meta": {"id": "WUA_STAT_25K_016472", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=156.79, s=33.99, n=14, compute CI."}]}, "response": "95% CI ≈ [138.98, 174.60].", "meta": {"id": "WUA_STAT_25K_016473", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.6, s=9.72, n=125, compute CI."}]}, "response": "95% CI ≈ [86.90, 90.30].", "meta": {"id": "WUA_STAT_25K_016474", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.78, compute E and Var."}]}, "response": "E[X]=14.82; Var[X]=3.260.", "meta": {"id": "WUA_STAT_25K_016475", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.07, μ=77.77, σ=4.92, compute z."}]}, "response": "z = -3.191.", "meta": {"id": "WUA_STAT_25K_016476", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.9, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.419.", "meta": {"id": "WUA_STAT_25K_016477", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=182.72, s=10.07, n=72, compute CI."}]}, "response": "95% CI ≈ [180.39, 185.05].", "meta": {"id": "WUA_STAT_25K_016478", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.48, μ=27.92, σ=4.45, compute z."}]}, "response": "z = 3.721.", "meta": {"id": "WUA_STAT_25K_016479", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.76, μ=3.06, σ=11.35, compute z."}]}, "response": "z = 0.767.", "meta": {"id": "WUA_STAT_25K_016480", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.05, compute E and Var."}]}, "response": "E[X]=1.75; Var[X]=1.662.", "meta": {"id": "WUA_STAT_25K_016481", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.75, μ=0.41, σ=1.63, compute z."}]}, "response": "z = 50.515.", "meta": {"id": "WUA_STAT_25K_016482", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=12.78, s=27.83, n=95, compute CI."}]}, "response": "95% CI ≈ [7.18, 18.38].", "meta": {"id": "WUA_STAT_25K_016483", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.108, sens=0.88, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.317.", "meta": {"id": "WUA_STAT_25K_016484", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.128, sens=0.72, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.413.", "meta": {"id": "WUA_STAT_25K_016485", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.2, sens=0.82, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.651.", "meta": {"id": "WUA_STAT_25K_016486", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.73, μ=58.66, σ=22.68, compute z."}]}, "response": "z = 1.370.", "meta": {"id": "WUA_STAT_25K_016487", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.207, sens=0.76, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.768.", "meta": {"id": "WUA_STAT_25K_016488", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.082, sens=0.84, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.231.", "meta": {"id": "WUA_STAT_25K_016489", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.62, compute E and Var."}]}, "response": "E[X]=13.64; Var[X]=5.183.", "meta": {"id": "WUA_STAT_25K_016490", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.87, μ=20.9, σ=13.25, compute z."}]}, "response": "z = 4.979.", "meta": {"id": "WUA_STAT_25K_016491", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.029, sens=0.73, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.195.", "meta": {"id": "WUA_STAT_25K_016492", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.2, μ=72.87, σ=15.87, compute z."}]}, "response": "z = -4.201.", "meta": {"id": "WUA_STAT_25K_016493", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.39, μ=17.06, σ=4.87, compute z."}]}, "response": "z = 8.076.", "meta": {"id": "WUA_STAT_25K_016494", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=19.87, s=18.14, n=101, compute CI."}]}, "response": "95% CI ≈ [16.33, 23.41].", "meta": {"id": "WUA_STAT_25K_016495", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.128, sens=0.82, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.546.", "meta": {"id": "WUA_STAT_25K_016496", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.4, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=4.320.", "meta": {"id": "WUA_STAT_25K_016497", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=137.74, s=4.53, n=20, compute CI."}]}, "response": "95% CI ≈ [135.75, 139.73].", "meta": {"id": "WUA_STAT_25K_016498", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.72, compute E and Var."}]}, "response": "E[X]=24.48; Var[X]=6.854.", "meta": {"id": "WUA_STAT_25K_016499", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.71, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.165.", "meta": {"id": "WUA_STAT_25K_016500", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=23.44, μ=6.29, σ=17.06, compute z."}]}, "response": "z = 1.005.", "meta": {"id": "WUA_STAT_25K_016501", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=87.6, μ=27.82, σ=16.57, compute z."}]}, "response": "z = 3.608.", "meta": {"id": "WUA_STAT_25K_016502", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.69, s=38.0, n=141, compute CI."}]}, "response": "95% CI ≈ [23.42, 35.96].", "meta": {"id": "WUA_STAT_25K_016503", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.77, μ=69.26, σ=13.23, compute z."}]}, "response": "z = -1.776.", "meta": {"id": "WUA_STAT_25K_016504", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=98.63, μ=97.92, σ=19.86, compute z."}]}, "response": "z = 0.036.", "meta": {"id": "WUA_STAT_25K_016505", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.117, sens=0.95, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.326.", "meta": {"id": "WUA_STAT_25K_016506", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.83, s=13.99, n=188, compute CI."}]}, "response": "95% CI ≈ [96.83, 100.83].", "meta": {"id": "WUA_STAT_25K_016507", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.55, compute E and Var."}]}, "response": "E[X]=10.45; Var[X]=4.702.", "meta": {"id": "WUA_STAT_25K_016508", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.131, sens=0.97, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.880.", "meta": {"id": "WUA_STAT_25K_016509", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.47, compute E and Var."}]}, "response": "E[X]=15.51; Var[X]=8.220.", "meta": {"id": "WUA_STAT_25K_016510", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.061, sens=0.89, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.224.", "meta": {"id": "WUA_STAT_25K_016511", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.08, compute E and Var."}]}, "response": "E[X]=1.76; Var[X]=1.619.", "meta": {"id": "WUA_STAT_25K_016512", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=107.57, s=7.67, n=95, compute CI."}]}, "response": "95% CI ≈ [106.03, 109.11].", "meta": {"id": "WUA_STAT_25K_016513", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.249, sens=0.89, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.663.", "meta": {"id": "WUA_STAT_25K_016514", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.57, μ=78.84, σ=7.51, compute z."}]}, "response": "z = -9.357.", "meta": {"id": "WUA_STAT_25K_016515", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.179, sens=0.92, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.445.", "meta": {"id": "WUA_STAT_25K_016516", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.28, compute E and Var."}]}, "response": "E[X]=5.60; Var[X]=4.032.", "meta": {"id": "WUA_STAT_25K_016517", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.32, compute E and Var."}]}, "response": "E[X]=1.60; Var[X]=1.088.", "meta": {"id": "WUA_STAT_25K_016518", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.79, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.452.", "meta": {"id": "WUA_STAT_25K_016519", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.045, sens=0.96, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.531.", "meta": {"id": "WUA_STAT_25K_016520", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.79, μ=37.06, σ=19.13, compute z."}]}, "response": "z = 0.090.", "meta": {"id": "WUA_STAT_25K_016521", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=147.73, s=24.57, n=177, compute CI."}]}, "response": "95% CI ≈ [144.11, 151.35].", "meta": {"id": "WUA_STAT_25K_016522", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.22, compute E and Var."}]}, "response": "E[X]=7.70; Var[X]=6.006.", "meta": {"id": "WUA_STAT_25K_016523", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.51, μ=10.08, σ=5.51, compute z."}]}, "response": "z = 9.152.", "meta": {"id": "WUA_STAT_25K_016524", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.23, μ=22.22, σ=23.6, compute z."}]}, "response": "z = 2.458.", "meta": {"id": "WUA_STAT_25K_016525", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.69, s=28.98, n=164, compute CI."}]}, "response": "95% CI ≈ [150.25, 159.13].", "meta": {"id": "WUA_STAT_25K_016526", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=99.92, s=20.44, n=189, compute CI."}]}, "response": "95% CI ≈ [97.01, 102.83].", "meta": {"id": "WUA_STAT_25K_016527", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.142, sens=0.85, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.520.", "meta": {"id": "WUA_STAT_25K_016528", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.46, compute E and Var."}]}, "response": "E[X]=11.96; Var[X]=6.458.", "meta": {"id": "WUA_STAT_25K_016529", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.48, compute E and Var."}]}, "response": "E[X]=12.96; Var[X]=6.739.", "meta": {"id": "WUA_STAT_25K_016530", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.09, μ=95.11, σ=21.73, compute z."}]}, "response": "z = -2.624.", "meta": {"id": "WUA_STAT_25K_016531", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.062, sens=0.94, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.509.", "meta": {"id": "WUA_STAT_25K_016532", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.098, sens=0.93, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.558.", "meta": {"id": "WUA_STAT_25K_016533", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.194, sens=0.95, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.638.", "meta": {"id": "WUA_STAT_25K_016534", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.143, sens=0.79, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.364.", "meta": {"id": "WUA_STAT_25K_016535", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=50.62, s=35.48, n=122, compute CI."}]}, "response": "95% CI ≈ [44.32, 56.92].", "meta": {"id": "WUA_STAT_25K_016536", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=84.51, μ=6.81, σ=6.15, compute z."}]}, "response": "z = 12.634.", "meta": {"id": "WUA_STAT_25K_016537", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=190.33, s=22.17, n=97, compute CI."}]}, "response": "95% CI ≈ [185.92, 194.74].", "meta": {"id": "WUA_STAT_25K_016538", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.79, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.741.", "meta": {"id": "WUA_STAT_25K_016539", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.2, μ=11.83, σ=7.02, compute z."}]}, "response": "z = 9.027.", "meta": {"id": "WUA_STAT_25K_016540", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=32.35, μ=2.08, σ=9.73, compute z."}]}, "response": "z = 3.111.", "meta": {"id": "WUA_STAT_25K_016541", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=79.69, μ=44.31, σ=11.76, compute z."}]}, "response": "z = 3.009.", "meta": {"id": "WUA_STAT_25K_016542", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=30.66, s=23.63, n=35, compute CI."}]}, "response": "95% CI ≈ [22.83, 38.49].", "meta": {"id": "WUA_STAT_25K_016543", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=59.34, μ=62.82, σ=23.93, compute z."}]}, "response": "z = -0.145.", "meta": {"id": "WUA_STAT_25K_016544", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.39, compute E and Var."}]}, "response": "E[X]=13.26; Var[X]=8.089.", "meta": {"id": "WUA_STAT_25K_016545", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.26, compute E and Var."}]}, "response": "E[X]=5.98; Var[X]=4.425.", "meta": {"id": "WUA_STAT_25K_016546", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=91.14, s=25.12, n=104, compute CI."}]}, "response": "95% CI ≈ [86.31, 95.97].", "meta": {"id": "WUA_STAT_25K_016547", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.2, sens=0.92, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.639.", "meta": {"id": "WUA_STAT_25K_016548", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=149.62, s=3.41, n=173, compute CI."}]}, "response": "95% CI ≈ [149.11, 150.13].", "meta": {"id": "WUA_STAT_25K_016549", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.136, sens=0.77, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.602.", "meta": {"id": "WUA_STAT_25K_016550", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.114, sens=0.9, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.405.", "meta": {"id": "WUA_STAT_25K_016551", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.038, sens=0.83, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.105.", "meta": {"id": "WUA_STAT_25K_016552", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.67, compute E and Var."}]}, "response": "E[X]=26.13; Var[X]=8.623.", "meta": {"id": "WUA_STAT_25K_016553", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.71, compute E and Var."}]}, "response": "E[X]=11.36; Var[X]=3.294.", "meta": {"id": "WUA_STAT_25K_016554", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.06, compute E and Var."}]}, "response": "E[X]=1.68; Var[X]=1.579.", "meta": {"id": "WUA_STAT_25K_016555", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.201, sens=0.71, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.373.", "meta": {"id": "WUA_STAT_25K_016556", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.73, μ=18.92, σ=17.97, compute z."}]}, "response": "z = 2.549.", "meta": {"id": "WUA_STAT_25K_016557", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.2, μ=36.27, σ=20.52, compute z."}]}, "response": "z = -0.442.", "meta": {"id": "WUA_STAT_25K_016558", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.73, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.627.", "meta": {"id": "WUA_STAT_25K_016559", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=68.86, μ=15.91, σ=16.37, compute z."}]}, "response": "z = 3.235.", "meta": {"id": "WUA_STAT_25K_016560", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=53.96, μ=90.9, σ=1.43, compute z."}]}, "response": "z = -25.832.", "meta": {"id": "WUA_STAT_25K_016561", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.95, μ=5.04, σ=4.28, compute z."}]}, "response": "z = 9.558.", "meta": {"id": "WUA_STAT_25K_016562", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=187.04, s=13.64, n=101, compute CI."}]}, "response": "95% CI ≈ [184.38, 189.70].", "meta": {"id": "WUA_STAT_25K_016563", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.03, s=35.66, n=152, compute CI."}]}, "response": "95% CI ≈ [168.36, 179.70].", "meta": {"id": "WUA_STAT_25K_016564", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=56.03, μ=77.26, σ=11.67, compute z."}]}, "response": "z = -1.819.", "meta": {"id": "WUA_STAT_25K_016565", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.026, sens=0.85, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.102.", "meta": {"id": "WUA_STAT_25K_016566", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.18, compute E and Var."}]}, "response": "E[X]=1.80; Var[X]=1.476.", "meta": {"id": "WUA_STAT_25K_016567", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=15.16, μ=81.73, σ=2.94, compute z."}]}, "response": "z = -22.643.", "meta": {"id": "WUA_STAT_25K_016568", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.01, s=10.23, n=123, compute CI."}]}, "response": "95% CI ≈ [140.20, 143.82].", "meta": {"id": "WUA_STAT_25K_016569", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=110.67, s=24.76, n=152, compute CI."}]}, "response": "95% CI ≈ [106.73, 114.61].", "meta": {"id": "WUA_STAT_25K_016570", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=19.0, s=36.14, n=182, compute CI."}]}, "response": "95% CI ≈ [13.75, 24.25].", "meta": {"id": "WUA_STAT_25K_016571", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.38, μ=64.3, σ=18.04, compute z."}]}, "response": "z = 1.446.", "meta": {"id": "WUA_STAT_25K_016572", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.17, s=32.54, n=137, compute CI."}]}, "response": "95% CI ≈ [174.72, 185.62].", "meta": {"id": "WUA_STAT_25K_016573", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.28, compute E and Var."}]}, "response": "E[X]=10.08; Var[X]=7.258.", "meta": {"id": "WUA_STAT_25K_016574", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=87.93, s=17.89, n=28, compute CI."}]}, "response": "95% CI ≈ [81.30, 94.56].", "meta": {"id": "WUA_STAT_25K_016575", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=47.62, s=25.49, n=113, compute CI."}]}, "response": "95% CI ≈ [42.92, 52.32].", "meta": {"id": "WUA_STAT_25K_016576", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.5, compute E and Var."}]}, "response": "E[X]=19.50; Var[X]=9.750.", "meta": {"id": "WUA_STAT_25K_016577", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=12.27, μ=50.28, σ=18.08, compute z."}]}, "response": "z = -2.102.", "meta": {"id": "WUA_STAT_25K_016578", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=61.07, μ=13.94, σ=13.14, compute z."}]}, "response": "z = 3.587.", "meta": {"id": "WUA_STAT_25K_016579", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.148, sens=0.95, spec=0.77. Compute PPV."}]}, "response": "PPV ≈ 0.418.", "meta": {"id": "WUA_STAT_25K_016580", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.31, compute E and Var."}]}, "response": "E[X]=6.82; Var[X]=4.706.", "meta": {"id": "WUA_STAT_25K_016581", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=199.83, s=2.66, n=19, compute CI."}]}, "response": "95% CI ≈ [198.63, 201.03].", "meta": {"id": "WUA_STAT_25K_016582", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=79.14, s=39.46, n=93, compute CI."}]}, "response": "95% CI ≈ [71.12, 87.16].", "meta": {"id": "WUA_STAT_25K_016583", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.2, sens=0.71, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.511.", "meta": {"id": "WUA_STAT_25K_016584", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=103.83, s=33.89, n=35, compute CI."}]}, "response": "95% CI ≈ [92.60, 115.06].", "meta": {"id": "WUA_STAT_25K_016585", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.06, s=33.46, n=20, compute CI."}]}, "response": "95% CI ≈ [60.40, 89.72].", "meta": {"id": "WUA_STAT_25K_016586", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.2, s=16.41, n=144, compute CI."}]}, "response": "95% CI ≈ [169.52, 174.88].", "meta": {"id": "WUA_STAT_25K_016587", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=45.46, μ=1.41, σ=17.03, compute z."}]}, "response": "z = 2.587.", "meta": {"id": "WUA_STAT_25K_016588", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.26, compute E and Var."}]}, "response": "E[X]=5.46; Var[X]=4.040.", "meta": {"id": "WUA_STAT_25K_016589", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.043, sens=0.71, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.615.", "meta": {"id": "WUA_STAT_25K_016590", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.177, sens=0.95, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.431.", "meta": {"id": "WUA_STAT_25K_016591", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=81.08, s=16.57, n=26, compute CI."}]}, "response": "95% CI ≈ [74.71, 87.45].", "meta": {"id": "WUA_STAT_25K_016592", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.19, compute E and Var."}]}, "response": "E[X]=5.32; Var[X]=4.309.", "meta": {"id": "WUA_STAT_25K_016593", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.29, compute E and Var."}]}, "response": "E[X]=6.38; Var[X]=4.530.", "meta": {"id": "WUA_STAT_25K_016594", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.069, sens=0.93, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.775.", "meta": {"id": "WUA_STAT_25K_016595", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.48, s=29.9, n=37, compute CI."}]}, "response": "95% CI ≈ [101.85, 121.11].", "meta": {"id": "WUA_STAT_25K_016596", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.54, s=39.95, n=100, compute CI."}]}, "response": "95% CI ≈ [3.71, 19.37].", "meta": {"id": "WUA_STAT_25K_016597", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.13, compute E and Var."}]}, "response": "E[X]=2.34; Var[X]=2.036.", "meta": {"id": "WUA_STAT_25K_016598", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.19, μ=39.94, σ=6.95, compute z."}]}, "response": "z = 5.504.", "meta": {"id": "WUA_STAT_25K_016599", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.13, compute E and Var."}]}, "response": "E[X]=2.73; Var[X]=2.375.", "meta": {"id": "WUA_STAT_25K_016600", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.82, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.737.", "meta": {"id": "WUA_STAT_25K_016601", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.91, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.911.", "meta": {"id": "WUA_STAT_25K_016602", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.15, compute E and Var."}]}, "response": "E[X]=2.70; Var[X]=2.295.", "meta": {"id": "WUA_STAT_25K_016603", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.81, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.706.", "meta": {"id": "WUA_STAT_25K_016604", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=178.3, s=10.31, n=176, compute CI."}]}, "response": "95% CI ≈ [176.78, 179.82].", "meta": {"id": "WUA_STAT_25K_016605", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.02, sens=0.81, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.084.", "meta": {"id": "WUA_STAT_25K_016606", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.51, compute E and Var."}]}, "response": "E[X]=7.65; Var[X]=3.748.", "meta": {"id": "WUA_STAT_25K_016607", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=33, p=0.55, compute E and Var."}]}, "response": "E[X]=18.15; Var[X]=8.168.", "meta": {"id": "WUA_STAT_25K_016608", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.051, sens=0.86, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.339.", "meta": {"id": "WUA_STAT_25K_016609", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.73, s=8.14, n=22, compute CI."}]}, "response": "95% CI ≈ [147.33, 154.13].", "meta": {"id": "WUA_STAT_25K_016610", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=179.65, s=8.38, n=154, compute CI."}]}, "response": "95% CI ≈ [178.33, 180.97].", "meta": {"id": "WUA_STAT_25K_016611", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.06, compute E and Var."}]}, "response": "E[X]=0.30; Var[X]=0.282.", "meta": {"id": "WUA_STAT_25K_016612", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=98.68, s=11.0, n=195, compute CI."}]}, "response": "95% CI ≈ [97.14, 100.22].", "meta": {"id": "WUA_STAT_25K_016613", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=128.29, s=6.97, n=122, compute CI."}]}, "response": "95% CI ≈ [127.05, 129.53].", "meta": {"id": "WUA_STAT_25K_016614", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.04, sens=0.92, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.299.", "meta": {"id": "WUA_STAT_25K_016615", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.23, sens=0.73, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.609.", "meta": {"id": "WUA_STAT_25K_016616", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.64, compute E and Var."}]}, "response": "E[X]=21.76; Var[X]=7.834.", "meta": {"id": "WUA_STAT_25K_016617", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=194.73, s=12.25, n=164, compute CI."}]}, "response": "95% CI ≈ [192.86, 196.60].", "meta": {"id": "WUA_STAT_25K_016618", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=92.57, μ=59.93, σ=15.14, compute z."}]}, "response": "z = 2.156.", "meta": {"id": "WUA_STAT_25K_016619", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.4, compute E and Var."}]}, "response": "E[X]=7.20; Var[X]=4.320.", "meta": {"id": "WUA_STAT_25K_016620", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.79, compute E and Var."}]}, "response": "E[X]=24.49; Var[X]=5.143.", "meta": {"id": "WUA_STAT_25K_016621", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.246, sens=0.86, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.596.", "meta": {"id": "WUA_STAT_25K_016622", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.183, sens=0.76, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.549.", "meta": {"id": "WUA_STAT_25K_016623", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=63.47, μ=81.45, σ=5.94, compute z."}]}, "response": "z = -3.027.", "meta": {"id": "WUA_STAT_25K_016624", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=57.68, s=37.14, n=75, compute CI."}]}, "response": "95% CI ≈ [49.27, 66.09].", "meta": {"id": "WUA_STAT_25K_016625", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.13, μ=95.4, σ=4.96, compute z."}]}, "response": "z = 0.147.", "meta": {"id": "WUA_STAT_25K_016626", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.73, compute E and Var."}]}, "response": "E[X]=13.14; Var[X]=3.548.", "meta": {"id": "WUA_STAT_25K_016627", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.78, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.343.", "meta": {"id": "WUA_STAT_25K_016628", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.63, compute E and Var."}]}, "response": "E[X]=25.20; Var[X]=9.324.", "meta": {"id": "WUA_STAT_25K_016629", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.67, μ=28.32, σ=3.29, compute z."}]}, "response": "z = -1.109.", "meta": {"id": "WUA_STAT_25K_016630", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=90.74, s=2.21, n=124, compute CI."}]}, "response": "95% CI ≈ [90.35, 91.13].", "meta": {"id": "WUA_STAT_25K_016631", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=31.77, s=21.08, n=15, compute CI."}]}, "response": "95% CI ≈ [21.10, 42.44].", "meta": {"id": "WUA_STAT_25K_016632", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.31, compute E and Var."}]}, "response": "E[X]=11.16; Var[X]=7.700.", "meta": {"id": "WUA_STAT_25K_016633", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.12, s=1.25, n=127, compute CI."}]}, "response": "95% CI ≈ [87.90, 88.34].", "meta": {"id": "WUA_STAT_25K_016634", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.43, compute E and Var."}]}, "response": "E[X]=12.47; Var[X]=7.108.", "meta": {"id": "WUA_STAT_25K_016635", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.85, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.162.", "meta": {"id": "WUA_STAT_25K_016636", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.13, s=11.17, n=51, compute CI."}]}, "response": "95% CI ≈ [195.06, 201.20].", "meta": {"id": "WUA_STAT_25K_016637", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.91, μ=85.93, σ=17.87, compute z."}]}, "response": "z = -2.519.", "meta": {"id": "WUA_STAT_25K_016638", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=35.06, μ=29.28, σ=9.34, compute z."}]}, "response": "z = 0.619.", "meta": {"id": "WUA_STAT_25K_016639", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.58, s=13.8, n=181, compute CI."}]}, "response": "95% CI ≈ [78.57, 82.59].", "meta": {"id": "WUA_STAT_25K_016640", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.88, s=14.17, n=26, compute CI."}]}, "response": "95% CI ≈ [169.43, 180.33].", "meta": {"id": "WUA_STAT_25K_016641", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.18, compute E and Var."}]}, "response": "E[X]=6.66; Var[X]=5.461.", "meta": {"id": "WUA_STAT_25K_016642", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.1, s=16.84, n=179, compute CI."}]}, "response": "95% CI ≈ [8.63, 13.57].", "meta": {"id": "WUA_STAT_25K_016643", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.173, sens=0.76, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.532.", "meta": {"id": "WUA_STAT_25K_016644", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.089, sens=0.77, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.429.", "meta": {"id": "WUA_STAT_25K_016645", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.37, compute E and Var."}]}, "response": "E[X]=12.95; Var[X]=8.159.", "meta": {"id": "WUA_STAT_25K_016646", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.78, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.558.", "meta": {"id": "WUA_STAT_25K_016647", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.071, sens=0.98, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.306.", "meta": {"id": "WUA_STAT_25K_016648", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.205, sens=0.88, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.636.", "meta": {"id": "WUA_STAT_25K_016649", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=11.45, μ=31.55, σ=11.67, compute z."}]}, "response": "z = -1.722.", "meta": {"id": "WUA_STAT_25K_016650", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.42, compute E and Var."}]}, "response": "E[X]=16.80; Var[X]=9.744.", "meta": {"id": "WUA_STAT_25K_016651", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.08, s=24.09, n=82, compute CI."}]}, "response": "95% CI ≈ [82.87, 93.29].", "meta": {"id": "WUA_STAT_25K_016652", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.133, sens=0.72, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.306.", "meta": {"id": "WUA_STAT_25K_016653", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=89.26, μ=0.83, σ=5.41, compute z."}]}, "response": "z = 16.346.", "meta": {"id": "WUA_STAT_25K_016654", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=55.31, μ=56.19, σ=4.83, compute z."}]}, "response": "z = -0.182.", "meta": {"id": "WUA_STAT_25K_016655", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.94, μ=34.47, σ=14.62, compute z."}]}, "response": "z = -0.652.", "meta": {"id": "WUA_STAT_25K_016656", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.228, sens=0.89, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.790.", "meta": {"id": "WUA_STAT_25K_016657", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.011, sens=0.94, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.058.", "meta": {"id": "WUA_STAT_25K_016658", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.79, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.949.", "meta": {"id": "WUA_STAT_25K_016659", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.203, sens=0.82, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.455.", "meta": {"id": "WUA_STAT_25K_016660", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.74, s=4.66, n=20, compute CI."}]}, "response": "95% CI ≈ [109.70, 113.78].", "meta": {"id": "WUA_STAT_25K_016661", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=123.44, s=34.94, n=148, compute CI."}]}, "response": "95% CI ≈ [117.81, 129.07].", "meta": {"id": "WUA_STAT_25K_016662", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.62, μ=22.4, σ=8.21, compute z."}]}, "response": "z = 5.143.", "meta": {"id": "WUA_STAT_25K_016663", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.228, sens=0.98, spec=0.85. Compute PPV."}]}, "response": "PPV ≈ 0.659.", "meta": {"id": "WUA_STAT_25K_016664", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=17.55, s=29.65, n=87, compute CI."}]}, "response": "95% CI ≈ [11.32, 23.78].", "meta": {"id": "WUA_STAT_25K_016665", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.237, sens=0.92, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.905.", "meta": {"id": "WUA_STAT_25K_016666", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.58, compute E and Var."}]}, "response": "E[X]=15.08; Var[X]=6.334.", "meta": {"id": "WUA_STAT_25K_016667", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=46.92, s=26.59, n=178, compute CI."}]}, "response": "95% CI ≈ [43.01, 50.83].", "meta": {"id": "WUA_STAT_25K_016668", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.159, sens=0.88, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.373.", "meta": {"id": "WUA_STAT_25K_016669", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.107, sens=0.74, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.297.", "meta": {"id": "WUA_STAT_25K_016670", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=21, p=0.21, compute E and Var."}]}, "response": "E[X]=4.41; Var[X]=3.484.", "meta": {"id": "WUA_STAT_25K_016671", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.46, compute E and Var."}]}, "response": "E[X]=2.30; Var[X]=1.242.", "meta": {"id": "WUA_STAT_25K_016672", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.41, compute E and Var."}]}, "response": "E[X]=15.99; Var[X]=9.434.", "meta": {"id": "WUA_STAT_25K_016673", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.11, μ=39.45, σ=22.75, compute z."}]}, "response": "z = -1.685.", "meta": {"id": "WUA_STAT_25K_016674", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=88.5, s=28.2, n=52, compute CI."}]}, "response": "95% CI ≈ [80.84, 96.16].", "meta": {"id": "WUA_STAT_25K_016675", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.68, compute E and Var."}]}, "response": "E[X]=20.40; Var[X]=6.528.", "meta": {"id": "WUA_STAT_25K_016676", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.139, sens=0.8, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.928.", "meta": {"id": "WUA_STAT_25K_016677", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=141.33, s=39.49, n=51, compute CI."}]}, "response": "95% CI ≈ [130.49, 152.17].", "meta": {"id": "WUA_STAT_25K_016678", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=33.07, μ=18.0, σ=20.4, compute z."}]}, "response": "z = 0.739.", "meta": {"id": "WUA_STAT_25K_016679", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=62.45, μ=31.7, σ=20.65, compute z."}]}, "response": "z = 1.489.", "meta": {"id": "WUA_STAT_25K_016680", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.3, μ=70.58, σ=19.41, compute z."}]}, "response": "z = -1.869.", "meta": {"id": "WUA_STAT_25K_016681", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=51.11, s=23.16, n=148, compute CI."}]}, "response": "95% CI ≈ [47.38, 54.84].", "meta": {"id": "WUA_STAT_25K_016682", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.022, sens=0.89, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.083.", "meta": {"id": "WUA_STAT_25K_016683", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.059, sens=0.81, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.149.", "meta": {"id": "WUA_STAT_25K_016684", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.48, s=21.86, n=57, compute CI."}]}, "response": "95% CI ≈ [17.80, 29.16].", "meta": {"id": "WUA_STAT_25K_016685", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=44.71, s=39.04, n=163, compute CI."}]}, "response": "95% CI ≈ [38.72, 50.70].", "meta": {"id": "WUA_STAT_25K_016686", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.09, sens=0.86, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.486.", "meta": {"id": "WUA_STAT_25K_016687", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.215, sens=0.95, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.565.", "meta": {"id": "WUA_STAT_25K_016688", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=91.84, s=9.88, n=159, compute CI."}]}, "response": "95% CI ≈ [90.30, 93.38].", "meta": {"id": "WUA_STAT_25K_016689", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.37, μ=36.19, σ=18.02, compute z."}]}, "response": "z = 0.398.", "meta": {"id": "WUA_STAT_25K_016690", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.07, sens=0.72, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.167.", "meta": {"id": "WUA_STAT_25K_016691", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=27.9, μ=81.84, σ=6.25, compute z."}]}, "response": "z = -8.630.", "meta": {"id": "WUA_STAT_25K_016692", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.75, μ=59.3, σ=10.33, compute z."}]}, "response": "z = -4.990.", "meta": {"id": "WUA_STAT_25K_016693", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.65, compute E and Var."}]}, "response": "E[X]=22.75; Var[X]=7.962.", "meta": {"id": "WUA_STAT_25K_016694", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.13, compute E and Var."}]}, "response": "E[X]=4.42; Var[X]=3.845.", "meta": {"id": "WUA_STAT_25K_016695", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.93, s=12.53, n=47, compute CI."}]}, "response": "95% CI ≈ [151.35, 158.51].", "meta": {"id": "WUA_STAT_25K_016696", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.29, compute E and Var."}]}, "response": "E[X]=2.90; Var[X]=2.059.", "meta": {"id": "WUA_STAT_25K_016697", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.95, μ=31.34, σ=2.68, compute z."}]}, "response": "z = -0.519.", "meta": {"id": "WUA_STAT_25K_016698", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.056, sens=0.81, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.286.", "meta": {"id": "WUA_STAT_25K_016699", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=66.88, s=11.46, n=124, compute CI."}]}, "response": "95% CI ≈ [64.86, 68.90].", "meta": {"id": "WUA_STAT_25K_016700", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.07, μ=47.77, σ=16.31, compute z."}]}, "response": "z = 1.551.", "meta": {"id": "WUA_STAT_25K_016701", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.68, s=14.39, n=134, compute CI."}]}, "response": "95% CI ≈ [128.24, 133.12].", "meta": {"id": "WUA_STAT_25K_016702", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.024, sens=0.99, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.125.", "meta": {"id": "WUA_STAT_25K_016703", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.126, sens=0.89, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.416.", "meta": {"id": "WUA_STAT_25K_016704", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=21.57, μ=83.57, σ=23.2, compute z."}]}, "response": "z = -2.672.", "meta": {"id": "WUA_STAT_25K_016705", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.171, sens=0.97, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.408.", "meta": {"id": "WUA_STAT_25K_016706", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=32, p=0.8, compute E and Var."}]}, "response": "E[X]=25.60; Var[X]=5.120.", "meta": {"id": "WUA_STAT_25K_016707", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.37, μ=21.99, σ=20.24, compute z."}]}, "response": "z = 1.303.", "meta": {"id": "WUA_STAT_25K_016708", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.49, compute E and Var."}]}, "response": "E[X]=12.25; Var[X]=6.248.", "meta": {"id": "WUA_STAT_25K_016709", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.84, s=1.47, n=22, compute CI."}]}, "response": "95% CI ≈ [62.23, 63.45].", "meta": {"id": "WUA_STAT_25K_016710", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.37, s=4.73, n=192, compute CI."}]}, "response": "95% CI ≈ [47.70, 49.04].", "meta": {"id": "WUA_STAT_25K_016711", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=73.4, s=12.56, n=35, compute CI."}]}, "response": "95% CI ≈ [69.24, 77.56].", "meta": {"id": "WUA_STAT_25K_016712", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.38, compute E and Var."}]}, "response": "E[X]=13.68; Var[X]=8.482.", "meta": {"id": "WUA_STAT_25K_016713", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.25, compute E and Var."}]}, "response": "E[X]=9.75; Var[X]=7.312.", "meta": {"id": "WUA_STAT_25K_016714", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=38.87, μ=69.76, σ=17.88, compute z."}]}, "response": "z = -1.728.", "meta": {"id": "WUA_STAT_25K_016715", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=67.96, μ=27.76, σ=15.21, compute z."}]}, "response": "z = 2.643.", "meta": {"id": "WUA_STAT_25K_016716", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.32, compute E and Var."}]}, "response": "E[X]=5.44; Var[X]=3.699.", "meta": {"id": "WUA_STAT_25K_016717", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=8.1, μ=61.75, σ=14.97, compute z."}]}, "response": "z = -3.584.", "meta": {"id": "WUA_STAT_25K_016718", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.48, compute E and Var."}]}, "response": "E[X]=3.36; Var[X]=1.747.", "meta": {"id": "WUA_STAT_25K_016719", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=75.28, s=6.3, n=187, compute CI."}]}, "response": "95% CI ≈ [74.38, 76.18].", "meta": {"id": "WUA_STAT_25K_016720", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.55, s=19.41, n=145, compute CI."}]}, "response": "95% CI ≈ [42.39, 48.71].", "meta": {"id": "WUA_STAT_25K_016721", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=35.96, s=14.32, n=12, compute CI."}]}, "response": "95% CI ≈ [27.86, 44.06].", "meta": {"id": "WUA_STAT_25K_016722", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.07, compute E and Var."}]}, "response": "E[X]=2.52; Var[X]=2.344.", "meta": {"id": "WUA_STAT_25K_016723", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.66, s=25.15, n=134, compute CI."}]}, "response": "95% CI ≈ [107.40, 115.92].", "meta": {"id": "WUA_STAT_25K_016724", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.27, μ=6.39, σ=21.51, compute z."}]}, "response": "z = -0.285.", "meta": {"id": "WUA_STAT_25K_016725", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.2, compute E and Var."}]}, "response": "E[X]=7.60; Var[X]=6.080.", "meta": {"id": "WUA_STAT_25K_016726", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.71, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.645.", "meta": {"id": "WUA_STAT_25K_016727", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.26, compute E and Var."}]}, "response": "E[X]=10.14; Var[X]=7.504.", "meta": {"id": "WUA_STAT_25K_016728", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.28, μ=11.28, σ=24.14, compute z."}]}, "response": "z = 3.563.", "meta": {"id": "WUA_STAT_25K_016729", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=88.22, μ=75.6, σ=11.54, compute z."}]}, "response": "z = 1.094.", "meta": {"id": "WUA_STAT_25K_016730", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.39, compute E and Var."}]}, "response": "E[X]=1.95; Var[X]=1.190.", "meta": {"id": "WUA_STAT_25K_016731", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.45, compute E and Var."}]}, "response": "E[X]=17.10; Var[X]=9.405.", "meta": {"id": "WUA_STAT_25K_016732", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.081, sens=0.84, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.215.", "meta": {"id": "WUA_STAT_25K_016733", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.95, μ=52.08, σ=11.06, compute z."}]}, "response": "z = -2.091.", "meta": {"id": "WUA_STAT_25K_016734", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=13.46, μ=37.48, σ=10.7, compute z."}]}, "response": "z = -2.245.", "meta": {"id": "WUA_STAT_25K_016735", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=80.41, s=32.85, n=157, compute CI."}]}, "response": "95% CI ≈ [75.27, 85.55].", "meta": {"id": "WUA_STAT_25K_016736", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=142.3, s=35.8, n=87, compute CI."}]}, "response": "95% CI ≈ [134.78, 149.82].", "meta": {"id": "WUA_STAT_25K_016737", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.48, compute E and Var."}]}, "response": "E[X]=17.76; Var[X]=9.235.", "meta": {"id": "WUA_STAT_25K_016738", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.37, compute E and Var."}]}, "response": "E[X]=5.18; Var[X]=3.263.", "meta": {"id": "WUA_STAT_25K_016739", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.29, compute E and Var."}]}, "response": "E[X]=7.54; Var[X]=5.353.", "meta": {"id": "WUA_STAT_25K_016740", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=60.58, μ=42.27, σ=6.74, compute z."}]}, "response": "z = 2.717.", "meta": {"id": "WUA_STAT_25K_016741", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.1, compute E and Var."}]}, "response": "E[X]=0.70; Var[X]=0.630.", "meta": {"id": "WUA_STAT_25K_016742", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.06, compute E and Var."}]}, "response": "E[X]=1.14; Var[X]=1.072.", "meta": {"id": "WUA_STAT_25K_016743", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.16, s=9.75, n=183, compute CI."}]}, "response": "95% CI ≈ [129.75, 132.57].", "meta": {"id": "WUA_STAT_25K_016744", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.41, s=30.47, n=28, compute CI."}]}, "response": "95% CI ≈ [37.12, 59.70].", "meta": {"id": "WUA_STAT_25K_016745", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.57, compute E and Var."}]}, "response": "E[X]=16.53; Var[X]=7.108.", "meta": {"id": "WUA_STAT_25K_016746", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.53, compute E and Var."}]}, "response": "E[X]=9.54; Var[X]=4.484.", "meta": {"id": "WUA_STAT_25K_016747", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.236, sens=0.94, spec=0.83. Compute PPV."}]}, "response": "PPV ≈ 0.631.", "meta": {"id": "WUA_STAT_25K_016748", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.41, compute E and Var."}]}, "response": "E[X]=2.87; Var[X]=1.693.", "meta": {"id": "WUA_STAT_25K_016749", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=26, p=0.29, compute E and Var."}]}, "response": "E[X]=7.54; Var[X]=5.353.", "meta": {"id": "WUA_STAT_25K_016750", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=188.32, s=32.34, n=56, compute CI."}]}, "response": "95% CI ≈ [179.85, 196.79].", "meta": {"id": "WUA_STAT_25K_016751", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.95, spec=0.76. Compute PPV."}]}, "response": "PPV ≈ 0.406.", "meta": {"id": "WUA_STAT_25K_016752", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.92, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.516.", "meta": {"id": "WUA_STAT_25K_016753", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=0.15, μ=7.85, σ=9.3, compute z."}]}, "response": "z = -0.828.", "meta": {"id": "WUA_STAT_25K_016754", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.44, μ=39.86, σ=19.22, compute z."}]}, "response": "z = -1.114.", "meta": {"id": "WUA_STAT_25K_016755", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.25, μ=39.29, σ=5.5, compute z."}]}, "response": "z = 0.902.", "meta": {"id": "WUA_STAT_25K_016756", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.34, μ=26.5, σ=24.33, compute z."}]}, "response": "z = 0.569.", "meta": {"id": "WUA_STAT_25K_016757", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.31, s=28.84, n=119, compute CI."}]}, "response": "95% CI ≈ [65.13, 75.49].", "meta": {"id": "WUA_STAT_25K_016758", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=6, p=0.76, compute E and Var."}]}, "response": "E[X]=4.56; Var[X]=1.094.", "meta": {"id": "WUA_STAT_25K_016759", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.15, s=20.01, n=166, compute CI."}]}, "response": "95% CI ≈ [86.11, 92.19].", "meta": {"id": "WUA_STAT_25K_016760", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.151, sens=0.81, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.445.", "meta": {"id": "WUA_STAT_25K_016761", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.242, sens=0.95, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.734.", "meta": {"id": "WUA_STAT_25K_016762", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.1, sens=0.99, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.297.", "meta": {"id": "WUA_STAT_25K_016763", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.46, compute E and Var."}]}, "response": "E[X]=13.80; Var[X]=7.452.", "meta": {"id": "WUA_STAT_25K_016764", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.61, s=29.42, n=198, compute CI."}]}, "response": "95% CI ≈ [88.51, 96.71].", "meta": {"id": "WUA_STAT_25K_016765", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.196, sens=0.99, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.889.", "meta": {"id": "WUA_STAT_25K_016766", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=119.09, s=28.94, n=164, compute CI."}]}, "response": "95% CI ≈ [114.66, 123.52].", "meta": {"id": "WUA_STAT_25K_016767", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.079, sens=0.74, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.202.", "meta": {"id": "WUA_STAT_25K_016768", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=154.71, s=24.24, n=29, compute CI."}]}, "response": "95% CI ≈ [145.89, 163.53].", "meta": {"id": "WUA_STAT_25K_016769", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.51, μ=95.21, σ=22.51, compute z."}]}, "response": "z = -3.896.", "meta": {"id": "WUA_STAT_25K_016770", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=143.11, s=19.56, n=106, compute CI."}]}, "response": "95% CI ≈ [139.39, 146.83].", "meta": {"id": "WUA_STAT_25K_016771", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=16.65, μ=82.96, σ=18.55, compute z."}]}, "response": "z = -3.575.", "meta": {"id": "WUA_STAT_25K_016772", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.89, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.520.", "meta": {"id": "WUA_STAT_25K_016773", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.81, μ=48.01, σ=6.01, compute z."}]}, "response": "z = 0.632.", "meta": {"id": "WUA_STAT_25K_016774", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.163, sens=0.95, spec=0.94. Compute PPV."}]}, "response": "PPV ≈ 0.755.", "meta": {"id": "WUA_STAT_25K_016775", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.59, compute E and Var."}]}, "response": "E[X]=15.93; Var[X]=6.531.", "meta": {"id": "WUA_STAT_25K_016776", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=85.85, μ=65.42, σ=18.13, compute z."}]}, "response": "z = 1.127.", "meta": {"id": "WUA_STAT_25K_016777", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=23.19, s=12.44, n=83, compute CI."}]}, "response": "95% CI ≈ [20.51, 25.87].", "meta": {"id": "WUA_STAT_25K_016778", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=9, p=0.3, compute E and Var."}]}, "response": "E[X]=2.70; Var[X]=1.890.", "meta": {"id": "WUA_STAT_25K_016779", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.84, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.190.", "meta": {"id": "WUA_STAT_25K_016780", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.16, compute E and Var."}]}, "response": "E[X]=3.68; Var[X]=3.091.", "meta": {"id": "WUA_STAT_25K_016781", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=135.38, s=2.64, n=18, compute CI."}]}, "response": "95% CI ≈ [134.16, 136.60].", "meta": {"id": "WUA_STAT_25K_016782", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.109, sens=0.81, spec=0.72. Compute PPV."}]}, "response": "PPV ≈ 0.261.", "meta": {"id": "WUA_STAT_25K_016783", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=25.73, s=20.12, n=143, compute CI."}]}, "response": "95% CI ≈ [22.43, 29.03].", "meta": {"id": "WUA_STAT_25K_016784", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.8, s=15.47, n=88, compute CI."}]}, "response": "95% CI ≈ [98.57, 105.03].", "meta": {"id": "WUA_STAT_25K_016785", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.58, compute E and Var."}]}, "response": "E[X]=11.02; Var[X]=4.628.", "meta": {"id": "WUA_STAT_25K_016786", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=82.69, μ=49.02, σ=2.44, compute z."}]}, "response": "z = 13.799.", "meta": {"id": "WUA_STAT_25K_016787", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.28, compute E and Var."}]}, "response": "E[X]=10.64; Var[X]=7.661.", "meta": {"id": "WUA_STAT_25K_016788", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.05, compute E and Var."}]}, "response": "E[X]=0.60; Var[X]=0.570.", "meta": {"id": "WUA_STAT_25K_016789", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.63, s=14.56, n=109, compute CI."}]}, "response": "95% CI ≈ [108.90, 114.36].", "meta": {"id": "WUA_STAT_25K_016790", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.74, compute E and Var."}]}, "response": "E[X]=14.06; Var[X]=3.656.", "meta": {"id": "WUA_STAT_25K_016791", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=81.15, μ=69.24, σ=18.64, compute z."}]}, "response": "z = 0.639.", "meta": {"id": "WUA_STAT_25K_016792", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.14, compute E and Var."}]}, "response": "E[X]=5.46; Var[X]=4.696.", "meta": {"id": "WUA_STAT_25K_016793", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=20, p=0.43, compute E and Var."}]}, "response": "E[X]=8.60; Var[X]=4.902.", "meta": {"id": "WUA_STAT_25K_016794", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.56, compute E and Var."}]}, "response": "E[X]=6.72; Var[X]=2.957.", "meta": {"id": "WUA_STAT_25K_016795", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.047, sens=0.77, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.352.", "meta": {"id": "WUA_STAT_25K_016796", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.164, sens=0.85, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.582.", "meta": {"id": "WUA_STAT_25K_016797", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.046, sens=0.92, spec=0.81. Compute PPV."}]}, "response": "PPV ≈ 0.189.", "meta": {"id": "WUA_STAT_25K_016798", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.71, compute E and Var."}]}, "response": "E[X]=24.85; Var[X]=7.207.", "meta": {"id": "WUA_STAT_25K_016799", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.64, s=1.48, n=86, compute CI."}]}, "response": "95% CI ≈ [175.33, 175.95].", "meta": {"id": "WUA_STAT_25K_016800", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.089, sens=0.72, spec=0.86. Compute PPV."}]}, "response": "PPV ≈ 0.334.", "meta": {"id": "WUA_STAT_25K_016801", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.58, s=17.78, n=73, compute CI."}]}, "response": "95% CI ≈ [146.50, 154.66].", "meta": {"id": "WUA_STAT_25K_016802", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.181, sens=0.84, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.537.", "meta": {"id": "WUA_STAT_25K_016803", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.62, compute E and Var."}]}, "response": "E[X]=7.44; Var[X]=2.827.", "meta": {"id": "WUA_STAT_25K_016804", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.19, sens=0.75, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.404.", "meta": {"id": "WUA_STAT_25K_016805", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.36, compute E and Var."}]}, "response": "E[X]=7.92; Var[X]=5.069.", "meta": {"id": "WUA_STAT_25K_016806", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.052, sens=0.86, spec=0.84. Compute PPV."}]}, "response": "PPV ≈ 0.228.", "meta": {"id": "WUA_STAT_25K_016807", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.75, compute E and Var."}]}, "response": "E[X]=7.50; Var[X]=1.875.", "meta": {"id": "WUA_STAT_25K_016808", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=131.6, s=8.51, n=196, compute CI."}]}, "response": "95% CI ≈ [130.41, 132.79].", "meta": {"id": "WUA_STAT_25K_016809", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=169.91, s=10.27, n=56, compute CI."}]}, "response": "95% CI ≈ [167.22, 172.60].", "meta": {"id": "WUA_STAT_25K_016810", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.15, compute E and Var."}]}, "response": "E[X]=5.70; Var[X]=4.845.", "meta": {"id": "WUA_STAT_25K_016811", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.118, sens=0.9, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.801.", "meta": {"id": "WUA_STAT_25K_016812", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=10, p=0.31, compute E and Var."}]}, "response": "E[X]=3.10; Var[X]=2.139.", "meta": {"id": "WUA_STAT_25K_016813", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=115.1, s=33.73, n=33, compute CI."}]}, "response": "95% CI ≈ [103.59, 126.61].", "meta": {"id": "WUA_STAT_25K_016814", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.199, sens=0.99, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.891.", "meta": {"id": "WUA_STAT_25K_016815", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=48.17, s=8.17, n=170, compute CI."}]}, "response": "95% CI ≈ [46.94, 49.40].", "meta": {"id": "WUA_STAT_25K_016816", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=70.82, s=32.87, n=19, compute CI."}]}, "response": "95% CI ≈ [56.04, 85.60].", "meta": {"id": "WUA_STAT_25K_016817", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.115, sens=0.8, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.342.", "meta": {"id": "WUA_STAT_25K_016818", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=109.51, s=13.46, n=179, compute CI."}]}, "response": "95% CI ≈ [107.54, 111.48].", "meta": {"id": "WUA_STAT_25K_016819", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.147, sens=0.71, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.710.", "meta": {"id": "WUA_STAT_25K_016820", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=6.48, μ=31.65, σ=20.76, compute z."}]}, "response": "z = -1.212.", "meta": {"id": "WUA_STAT_25K_016821", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.035, sens=0.91, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.320.", "meta": {"id": "WUA_STAT_25K_016822", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.53, μ=61.41, σ=14.32, compute z."}]}, "response": "z = -1.318.", "meta": {"id": "WUA_STAT_25K_016823", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.73, compute E and Var."}]}, "response": "E[X]=10.22; Var[X]=2.759.", "meta": {"id": "WUA_STAT_25K_016824", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.166, sens=0.9, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.579.", "meta": {"id": "WUA_STAT_25K_016825", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.89, μ=57.59, σ=24.34, compute z."}]}, "response": "z = 0.300.", "meta": {"id": "WUA_STAT_25K_016826", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.69, compute E and Var."}]}, "response": "E[X]=15.18; Var[X]=4.706.", "meta": {"id": "WUA_STAT_25K_016827", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.19, compute E and Var."}]}, "response": "E[X]=3.61; Var[X]=2.924.", "meta": {"id": "WUA_STAT_25K_016828", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=101.14, s=36.77, n=189, compute CI."}]}, "response": "95% CI ≈ [95.90, 106.38].", "meta": {"id": "WUA_STAT_25K_016829", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=72.17, μ=33.98, σ=8.08, compute z."}]}, "response": "z = 4.726.", "meta": {"id": "WUA_STAT_25K_016830", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=50.47, μ=18.88, σ=7.28, compute z."}]}, "response": "z = 4.339.", "meta": {"id": "WUA_STAT_25K_016831", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.083, sens=0.81, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.250.", "meta": {"id": "WUA_STAT_25K_016832", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=19, p=0.75, compute E and Var."}]}, "response": "E[X]=14.25; Var[X]=3.562.", "meta": {"id": "WUA_STAT_25K_016833", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.022, sens=0.7, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.344.", "meta": {"id": "WUA_STAT_25K_016834", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.74, μ=34.89, σ=17.31, compute z."}]}, "response": "z = 0.338.", "meta": {"id": "WUA_STAT_25K_016835", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=7.28, μ=57.9, σ=10.15, compute z."}]}, "response": "z = -4.987.", "meta": {"id": "WUA_STAT_25K_016836", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.049, sens=0.98, spec=0.89. Compute PPV."}]}, "response": "PPV ≈ 0.315.", "meta": {"id": "WUA_STAT_25K_016837", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.68, compute E and Var."}]}, "response": "E[X]=23.80; Var[X]=7.616.", "meta": {"id": "WUA_STAT_25K_016838", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=34.7, μ=77.63, σ=1.24, compute z."}]}, "response": "z = -34.621.", "meta": {"id": "WUA_STAT_25K_016839", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=28.43, μ=23.77, σ=16.19, compute z."}]}, "response": "z = 0.288.", "meta": {"id": "WUA_STAT_25K_016840", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=86.23, s=4.96, n=43, compute CI."}]}, "response": "95% CI ≈ [84.75, 87.71].", "meta": {"id": "WUA_STAT_25K_016841", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=28, p=0.22, compute E and Var."}]}, "response": "E[X]=6.16; Var[X]=4.805.", "meta": {"id": "WUA_STAT_25K_016842", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=37, p=0.59, compute E and Var."}]}, "response": "E[X]=21.83; Var[X]=8.950.", "meta": {"id": "WUA_STAT_25K_016843", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.96, μ=51.49, σ=21.7, compute z."}]}, "response": "z = -0.117.", "meta": {"id": "WUA_STAT_25K_016844", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=198.01, s=9.67, n=126, compute CI."}]}, "response": "95% CI ≈ [196.32, 199.70].", "meta": {"id": "WUA_STAT_25K_016845", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.038, sens=0.78, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.435.", "meta": {"id": "WUA_STAT_25K_016846", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.52, s=13.97, n=92, compute CI."}]}, "response": "95% CI ≈ [59.67, 65.37].", "meta": {"id": "WUA_STAT_25K_016847", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.04, μ=84.15, σ=22.74, compute z."}]}, "response": "z = -2.951.", "meta": {"id": "WUA_STAT_25K_016848", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.2, compute E and Var."}]}, "response": "E[X]=7.80; Var[X]=6.240.", "meta": {"id": "WUA_STAT_25K_016849", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.78, compute E and Var."}]}, "response": "E[X]=27.30; Var[X]=6.006.", "meta": {"id": "WUA_STAT_25K_016850", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.027, sens=0.74, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.066.", "meta": {"id": "WUA_STAT_25K_016851", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=51.68, μ=34.53, σ=22.49, compute z."}]}, "response": "z = 0.763.", "meta": {"id": "WUA_STAT_25K_016852", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.206, sens=0.78, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.438.", "meta": {"id": "WUA_STAT_25K_016853", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=111.96, s=22.61, n=71, compute CI."}]}, "response": "95% CI ≈ [106.70, 117.22].", "meta": {"id": "WUA_STAT_25K_016854", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.62, s=7.28, n=18, compute CI."}]}, "response": "95% CI ≈ [62.26, 68.98].", "meta": {"id": "WUA_STAT_25K_016855", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.8, μ=36.12, σ=15.35, compute z."}]}, "response": "z = -2.236.", "meta": {"id": "WUA_STAT_25K_016856", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=180.72, s=31.67, n=164, compute CI."}]}, "response": "95% CI ≈ [175.87, 185.57].", "meta": {"id": "WUA_STAT_25K_016857", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.16, sens=0.93, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.689.", "meta": {"id": "WUA_STAT_25K_016858", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.22, μ=3.59, σ=9.25, compute z."}]}, "response": "z = 7.528.", "meta": {"id": "WUA_STAT_25K_016859", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=31, p=0.22, compute E and Var."}]}, "response": "E[X]=6.82; Var[X]=5.320.", "meta": {"id": "WUA_STAT_25K_016860", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.28, μ=30.32, σ=8.94, compute z."}]}, "response": "z = 0.779.", "meta": {"id": "WUA_STAT_25K_016861", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.86, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.457.", "meta": {"id": "WUA_STAT_25K_016862", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.026, sens=0.97, spec=0.74. Compute PPV."}]}, "response": "PPV ≈ 0.091.", "meta": {"id": "WUA_STAT_25K_016863", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.113, sens=0.84, spec=0.79. Compute PPV."}]}, "response": "PPV ≈ 0.338.", "meta": {"id": "WUA_STAT_25K_016864", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=129.27, s=31.51, n=126, compute CI."}]}, "response": "95% CI ≈ [123.77, 134.77].", "meta": {"id": "WUA_STAT_25K_016865", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=39, p=0.2, compute E and Var."}]}, "response": "E[X]=7.80; Var[X]=6.240.", "meta": {"id": "WUA_STAT_25K_016866", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.17, compute E and Var."}]}, "response": "E[X]=2.55; Var[X]=2.117.", "meta": {"id": "WUA_STAT_25K_016867", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.31, compute E and Var."}]}, "response": "E[X]=8.37; Var[X]=5.775.", "meta": {"id": "WUA_STAT_25K_016868", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=52.57, s=28.05, n=82, compute CI."}]}, "response": "95% CI ≈ [46.50, 58.64].", "meta": {"id": "WUA_STAT_25K_016869", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=41.37, μ=88.38, σ=14.71, compute z."}]}, "response": "z = -3.196.", "meta": {"id": "WUA_STAT_25K_016870", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=58.28, μ=75.63, σ=20.97, compute z."}]}, "response": "z = -0.827.", "meta": {"id": "WUA_STAT_25K_016871", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=169.84, s=35.77, n=158, compute CI."}]}, "response": "95% CI ≈ [164.26, 175.42].", "meta": {"id": "WUA_STAT_25K_016872", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.04, μ=54.22, σ=11.43, compute z."}]}, "response": "z = -2.203.", "meta": {"id": "WUA_STAT_25K_016873", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=31.2, μ=9.79, σ=10.0, compute z."}]}, "response": "z = 2.141.", "meta": {"id": "WUA_STAT_25K_016874", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=130.01, s=14.2, n=169, compute CI."}]}, "response": "95% CI ≈ [127.87, 132.15].", "meta": {"id": "WUA_STAT_25K_016875", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=92.93, s=2.84, n=44, compute CI."}]}, "response": "95% CI ≈ [92.09, 93.77].", "meta": {"id": "WUA_STAT_25K_016876", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.074, sens=0.72, spec=0.75. Compute PPV."}]}, "response": "PPV ≈ 0.187.", "meta": {"id": "WUA_STAT_25K_016877", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=127.29, s=8.76, n=32, compute CI."}]}, "response": "95% CI ≈ [124.25, 130.33].", "meta": {"id": "WUA_STAT_25K_016878", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.189, sens=0.79, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.672.", "meta": {"id": "WUA_STAT_25K_016879", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=7, p=0.68, compute E and Var."}]}, "response": "E[X]=4.76; Var[X]=1.523.", "meta": {"id": "WUA_STAT_25K_016880", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=78.62, μ=25.39, σ=8.04, compute z."}]}, "response": "z = 6.621.", "meta": {"id": "WUA_STAT_25K_016881", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.13, compute E and Var."}]}, "response": "E[X]=0.65; Var[X]=0.566.", "meta": {"id": "WUA_STAT_25K_016882", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=172.48, s=36.94, n=200, compute CI."}]}, "response": "95% CI ≈ [167.36, 177.60].", "meta": {"id": "WUA_STAT_25K_016883", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.81, μ=97.6, σ=3.27, compute z."}]}, "response": "z = -29.294.", "meta": {"id": "WUA_STAT_25K_016884", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=12, p=0.4, compute E and Var."}]}, "response": "E[X]=4.80; Var[X]=2.880.", "meta": {"id": "WUA_STAT_25K_016885", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.24, compute E and Var."}]}, "response": "E[X]=3.36; Var[X]=2.554.", "meta": {"id": "WUA_STAT_25K_016886", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.086, sens=0.9, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.239.", "meta": {"id": "WUA_STAT_25K_016887", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.73, compute E and Var."}]}, "response": "E[X]=24.82; Var[X]=6.701.", "meta": {"id": "WUA_STAT_25K_016888", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=174.21, s=28.32, n=142, compute CI."}]}, "response": "95% CI ≈ [169.55, 178.87].", "meta": {"id": "WUA_STAT_25K_016889", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.202, sens=0.82, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.748.", "meta": {"id": "WUA_STAT_25K_016890", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=29.71, μ=94.07, σ=22.34, compute z."}]}, "response": "z = -2.881.", "meta": {"id": "WUA_STAT_25K_016891", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=33.55, s=31.93, n=194, compute CI."}]}, "response": "95% CI ≈ [29.06, 38.04].", "meta": {"id": "WUA_STAT_25K_016892", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=23, p=0.29, compute E and Var."}]}, "response": "E[X]=6.67; Var[X]=4.736.", "meta": {"id": "WUA_STAT_25K_016893", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=43.3, μ=66.74, σ=24.41, compute z."}]}, "response": "z = -0.960.", "meta": {"id": "WUA_STAT_25K_016894", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.198, sens=0.77, spec=0.73. Compute PPV."}]}, "response": "PPV ≈ 0.413.", "meta": {"id": "WUA_STAT_25K_016895", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.38, μ=46.03, σ=15.55, compute z."}]}, "response": "z = -0.106.", "meta": {"id": "WUA_STAT_25K_016896", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=38.76, s=2.67, n=65, compute CI."}]}, "response": "95% CI ≈ [38.11, 39.41].", "meta": {"id": "WUA_STAT_25K_016897", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=89.38, s=12.06, n=96, compute CI."}]}, "response": "95% CI ≈ [86.97, 91.79].", "meta": {"id": "WUA_STAT_25K_016898", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=103.81, s=36.3, n=69, compute CI."}]}, "response": "95% CI ≈ [95.24, 112.38].", "meta": {"id": "WUA_STAT_25K_016899", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=24.73, μ=78.77, σ=21.95, compute z."}]}, "response": "z = -2.462.", "meta": {"id": "WUA_STAT_25K_016900", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=11.41, s=16.45, n=89, compute CI."}]}, "response": "95% CI ≈ [7.99, 14.83].", "meta": {"id": "WUA_STAT_25K_016901", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.132, sens=0.72, spec=0.8. Compute PPV."}]}, "response": "PPV ≈ 0.354.", "meta": {"id": "WUA_STAT_25K_016902", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=37.51, μ=65.49, σ=8.22, compute z."}]}, "response": "z = -3.404.", "meta": {"id": "WUA_STAT_25K_016903", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=96.9, μ=16.95, σ=8.06, compute z."}]}, "response": "z = 9.919.", "meta": {"id": "WUA_STAT_25K_016904", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.039, sens=0.71, spec=0.92. Compute PPV."}]}, "response": "PPV ≈ 0.265.", "meta": {"id": "WUA_STAT_25K_016905", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.121, sens=0.88, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.482.", "meta": {"id": "WUA_STAT_25K_016906", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.91, spec=0.82. Compute PPV."}]}, "response": "PPV ≈ 0.365.", "meta": {"id": "WUA_STAT_25K_016907", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.53, compute E and Var."}]}, "response": "E[X]=2.65; Var[X]=1.246.", "meta": {"id": "WUA_STAT_25K_016908", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.152, sens=0.84, spec=0.93. Compute PPV."}]}, "response": "PPV ≈ 0.683.", "meta": {"id": "WUA_STAT_25K_016909", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.15, sens=0.91, spec=0.91. Compute PPV."}]}, "response": "PPV ≈ 0.641.", "meta": {"id": "WUA_STAT_25K_016910", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=150.64, s=1.83, n=150, compute CI."}]}, "response": "95% CI ≈ [150.35, 150.93].", "meta": {"id": "WUA_STAT_25K_016911", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=67.47, s=12.7, n=107, compute CI."}]}, "response": "95% CI ≈ [65.06, 69.88].", "meta": {"id": "WUA_STAT_25K_016912", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.03, μ=69.97, σ=15.77, compute z."}]}, "response": "z = 1.018.", "meta": {"id": "WUA_STAT_25K_016913", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=42.48, μ=31.81, σ=18.79, compute z."}]}, "response": "z = 0.568.", "meta": {"id": "WUA_STAT_25K_016914", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=29.17, s=18.12, n=80, compute CI."}]}, "response": "95% CI ≈ [25.20, 33.14].", "meta": {"id": "WUA_STAT_25K_016915", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.13, compute E and Var."}]}, "response": "E[X]=3.77; Var[X]=3.280.", "meta": {"id": "WUA_STAT_25K_016916", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=138.43, s=10.72, n=125, compute CI."}]}, "response": "95% CI ≈ [136.55, 140.31].", "meta": {"id": "WUA_STAT_25K_016917", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.102, sens=0.92, spec=0.97. Compute PPV."}]}, "response": "PPV ≈ 0.777.", "meta": {"id": "WUA_STAT_25K_016918", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=53.37, s=23.35, n=98, compute CI."}]}, "response": "95% CI ≈ [48.75, 57.99].", "meta": {"id": "WUA_STAT_25K_016919", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=18.66, μ=59.34, σ=2.03, compute z."}]}, "response": "z = -20.039.", "meta": {"id": "WUA_STAT_25K_016920", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=184.44, s=10.14, n=83, compute CI."}]}, "response": "95% CI ≈ [182.26, 186.62].", "meta": {"id": "WUA_STAT_25K_016921", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.44, compute E and Var."}]}, "response": "E[X]=7.04; Var[X]=3.942.", "meta": {"id": "WUA_STAT_25K_016922", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.42, compute E and Var."}]}, "response": "E[X]=7.14; Var[X]=4.141.", "meta": {"id": "WUA_STAT_25K_016923", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=34, p=0.07, compute E and Var."}]}, "response": "E[X]=2.38; Var[X]=2.213.", "meta": {"id": "WUA_STAT_25K_016924", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=90.69, μ=21.67, σ=2.13, compute z."}]}, "response": "z = 32.404.", "meta": {"id": "WUA_STAT_25K_016925", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=1.4, μ=90.16, σ=24.62, compute z."}]}, "response": "z = -3.605.", "meta": {"id": "WUA_STAT_25K_016926", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.69, compute E and Var."}]}, "response": "E[X]=10.35; Var[X]=3.209.", "meta": {"id": "WUA_STAT_25K_016927", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=18.95, s=10.18, n=88, compute CI."}]}, "response": "95% CI ≈ [16.82, 21.08].", "meta": {"id": "WUA_STAT_25K_016928", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=72.19, s=3.66, n=49, compute CI."}]}, "response": "95% CI ≈ [71.17, 73.21].", "meta": {"id": "WUA_STAT_25K_016929", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=17, p=0.07, compute E and Var."}]}, "response": "E[X]=1.19; Var[X]=1.107.", "meta": {"id": "WUA_STAT_25K_016930", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.216, sens=0.72, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.406.", "meta": {"id": "WUA_STAT_25K_016931", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=14, p=0.6, compute E and Var."}]}, "response": "E[X]=8.40; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_016932", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=11, p=0.44, compute E and Var."}]}, "response": "E[X]=4.84; Var[X]=2.710.", "meta": {"id": "WUA_STAT_25K_016933", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=73.68, μ=46.14, σ=11.47, compute z."}]}, "response": "z = 2.401.", "meta": {"id": "WUA_STAT_25K_016934", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=40, p=0.19, compute E and Var."}]}, "response": "E[X]=7.60; Var[X]=6.156.", "meta": {"id": "WUA_STAT_25K_016935", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=44.77, μ=76.11, σ=7.78, compute z."}]}, "response": "z = -4.028.", "meta": {"id": "WUA_STAT_25K_016936", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.36, μ=92.49, σ=21.67, compute z."}]}, "response": "z = 0.225.", "meta": {"id": "WUA_STAT_25K_016937", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=62.01, s=36.92, n=73, compute CI."}]}, "response": "95% CI ≈ [53.54, 70.48].", "meta": {"id": "WUA_STAT_25K_016938", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=166.38, s=18.1, n=101, compute CI."}]}, "response": "95% CI ≈ [162.85, 169.91].", "meta": {"id": "WUA_STAT_25K_016939", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=76.8, μ=83.14, σ=8.82, compute z."}]}, "response": "z = -0.719.", "meta": {"id": "WUA_STAT_25K_016940", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=48.02, μ=53.63, σ=14.42, compute z."}]}, "response": "z = -0.389.", "meta": {"id": "WUA_STAT_25K_016941", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=77.56, s=20.85, n=73, compute CI."}]}, "response": "95% CI ≈ [72.78, 82.34].", "meta": {"id": "WUA_STAT_25K_016942", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=163.52, s=14.7, n=70, compute CI."}]}, "response": "95% CI ≈ [160.08, 166.96].", "meta": {"id": "WUA_STAT_25K_016943", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=148.12, s=15.16, n=57, compute CI."}]}, "response": "95% CI ≈ [144.18, 152.06].", "meta": {"id": "WUA_STAT_25K_016944", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=40.46, s=3.98, n=167, compute CI."}]}, "response": "95% CI ≈ [39.86, 41.06].", "meta": {"id": "WUA_STAT_25K_016945", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=52.24, μ=64.15, σ=16.28, compute z."}]}, "response": "z = -0.732.", "meta": {"id": "WUA_STAT_25K_016946", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=24, p=0.5, compute E and Var."}]}, "response": "E[X]=12.00; Var[X]=6.000.", "meta": {"id": "WUA_STAT_25K_016947", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=30, p=0.56, compute E and Var."}]}, "response": "E[X]=16.80; Var[X]=7.392.", "meta": {"id": "WUA_STAT_25K_016948", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.155, sens=0.93, spec=0.95. Compute PPV."}]}, "response": "PPV ≈ 0.773.", "meta": {"id": "WUA_STAT_25K_016949", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=136.63, s=30.6, n=11, compute CI."}]}, "response": "95% CI ≈ [118.55, 154.71].", "meta": {"id": "WUA_STAT_25K_016950", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=83.04, s=11.51, n=122, compute CI."}]}, "response": "95% CI ≈ [81.00, 85.08].", "meta": {"id": "WUA_STAT_25K_016951", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.153, sens=0.97, spec=0.9. Compute PPV."}]}, "response": "PPV ≈ 0.637.", "meta": {"id": "WUA_STAT_25K_016952", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.22, sens=0.86, spec=0.96. Compute PPV."}]}, "response": "PPV ≈ 0.858.", "meta": {"id": "WUA_STAT_25K_016953", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=151.72, s=39.1, n=88, compute CI."}]}, "response": "95% CI ≈ [143.55, 159.89].", "meta": {"id": "WUA_STAT_25K_016954", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=9.43, μ=76.56, σ=8.43, compute z."}]}, "response": "z = -7.963.", "meta": {"id": "WUA_STAT_25K_016955", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=18, p=0.77, compute E and Var."}]}, "response": "E[X]=13.86; Var[X]=3.188.", "meta": {"id": "WUA_STAT_25K_016956", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=15, p=0.76, compute E and Var."}]}, "response": "E[X]=11.40; Var[X]=2.736.", "meta": {"id": "WUA_STAT_25K_016957", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=20.45, μ=61.55, σ=12.0, compute z."}]}, "response": "z = -3.425.", "meta": {"id": "WUA_STAT_25K_016958", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.131, sens=0.74, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.848.", "meta": {"id": "WUA_STAT_25K_016959", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.34, compute E and Var."}]}, "response": "E[X]=9.86; Var[X]=6.508.", "meta": {"id": "WUA_STAT_25K_016960", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=27, p=0.18, compute E and Var."}]}, "response": "E[X]=4.86; Var[X]=3.985.", "meta": {"id": "WUA_STAT_25K_016961", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=86.23, μ=13.7, σ=5.13, compute z."}]}, "response": "z = 14.138.", "meta": {"id": "WUA_STAT_25K_016962", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=175.17, s=15.26, n=137, compute CI."}]}, "response": "95% CI ≈ [172.61, 177.73].", "meta": {"id": "WUA_STAT_25K_016963", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.188, sens=0.9, spec=0.98. Compute PPV."}]}, "response": "PPV ≈ 0.912.", "meta": {"id": "WUA_STAT_25K_016964", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.127, sens=0.78, spec=0.87. Compute PPV."}]}, "response": "PPV ≈ 0.466.", "meta": {"id": "WUA_STAT_25K_016965", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=5, p=0.73, compute E and Var."}]}, "response": "E[X]=3.65; Var[X]=0.986.", "meta": {"id": "WUA_STAT_25K_016966", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=19.35, s=10.68, n=94, compute CI."}]}, "response": "95% CI ≈ [17.19, 21.51].", "meta": {"id": "WUA_STAT_25K_016967", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=19.91, μ=26.41, σ=18.61, compute z."}]}, "response": "z = -0.349.", "meta": {"id": "WUA_STAT_25K_016968", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.078, sens=0.74, spec=0.7. Compute PPV."}]}, "response": "PPV ≈ 0.173.", "meta": {"id": "WUA_STAT_25K_016969", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.146, sens=0.7, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.352.", "meta": {"id": "WUA_STAT_25K_016970", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=80.74, μ=17.36, σ=16.63, compute z."}]}, "response": "z = 3.811.", "meta": {"id": "WUA_STAT_25K_016971", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=185.93, s=14.89, n=169, compute CI."}]}, "response": "95% CI ≈ [183.69, 188.17].", "meta": {"id": "WUA_STAT_25K_016972", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=148.05, s=16.84, n=191, compute CI."}]}, "response": "95% CI ≈ [145.66, 150.44].", "meta": {"id": "WUA_STAT_25K_016973", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.69, μ=99.04, σ=9.1, compute z."}]}, "response": "z = -5.423.", "meta": {"id": "WUA_STAT_25K_016974", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.22, compute E and Var."}]}, "response": "E[X]=7.70; Var[X]=6.006.", "meta": {"id": "WUA_STAT_25K_016975", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=45.35, s=9.53, n=33, compute CI."}]}, "response": "95% CI ≈ [42.10, 48.60].", "meta": {"id": "WUA_STAT_25K_016976", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=75.39, μ=99.27, σ=21.11, compute z."}]}, "response": "z = -1.131.", "meta": {"id": "WUA_STAT_25K_016977", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.099, sens=0.73, spec=0.71. Compute PPV."}]}, "response": "PPV ≈ 0.217.", "meta": {"id": "WUA_STAT_25K_016978", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=128.33, s=29.3, n=116, compute CI."}]}, "response": "95% CI ≈ [123.00, 133.66].", "meta": {"id": "WUA_STAT_25K_016979", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=82.25, s=10.43, n=71, compute CI."}]}, "response": "95% CI ≈ [79.82, 84.68].", "meta": {"id": "WUA_STAT_25K_016980", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=15.68, s=17.59, n=145, compute CI."}]}, "response": "95% CI ≈ [12.82, 18.54].", "meta": {"id": "WUA_STAT_25K_016981", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=64.6, μ=48.4, σ=5.43, compute z."}]}, "response": "z = 2.983.", "meta": {"id": "WUA_STAT_25K_016982", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.66, μ=6.15, σ=21.78, compute z."}]}, "response": "z = 1.584.", "meta": {"id": "WUA_STAT_25K_016983", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=25, p=0.4, compute E and Var."}]}, "response": "E[X]=10.00; Var[X]=6.000.", "meta": {"id": "WUA_STAT_25K_016984", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.21, sens=0.72, spec=0.99. Compute PPV."}]}, "response": "PPV ≈ 0.950.", "meta": {"id": "WUA_STAT_25K_016985", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=16, p=0.7, compute E and Var."}]}, "response": "E[X]=11.20; Var[X]=3.360.", "meta": {"id": "WUA_STAT_25K_016986", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.071, sens=0.74, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.204.", "meta": {"id": "WUA_STAT_25K_016987", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=29, p=0.11, compute E and Var."}]}, "response": "E[X]=3.19; Var[X]=2.839.", "meta": {"id": "WUA_STAT_25K_016988", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=17.12, μ=28.9, σ=6.46, compute z."}]}, "response": "z = -1.824.", "meta": {"id": "WUA_STAT_25K_016989", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=36, p=0.23, compute E and Var."}]}, "response": "E[X]=8.28; Var[X]=6.376.", "meta": {"id": "WUA_STAT_25K_016990", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.235, sens=0.88, spec=0.88. Compute PPV."}]}, "response": "PPV ≈ 0.693.", "meta": {"id": "WUA_STAT_25K_016991", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "95% CI (z-approx): x̄ ± 1.96*(s/√n). Given x̄=65.42, s=1.91, n=182, compute CI."}]}, "response": "95% CI ≈ [65.14, 65.70].", "meta": {"id": "WUA_STAT_25K_016992", "domain": "master_scholars_academics", "subdomain": "stats_math::confidence_intervals", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=8, p=0.41, compute E and Var."}]}, "response": "E[X]=3.28; Var[X]=1.935.", "meta": {"id": "WUA_STAT_25K_016993", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=22, p=0.7, compute E and Var."}]}, "response": "E[X]=15.40; Var[X]=4.620.", "meta": {"id": "WUA_STAT_25K_016994", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=38, p=0.17, compute E and Var."}]}, "response": "E[X]=6.46; Var[X]=5.362.", "meta": {"id": "WUA_STAT_25K_016995", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=40.4, μ=91.65, σ=6.27, compute z."}]}, "response": "z = -8.174.", "meta": {"id": "WUA_STAT_25K_016996", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=49.96, μ=68.51, σ=11.09, compute z."}]}, "response": "z = -1.673.", "meta": {"id": "WUA_STAT_25K_016997", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "PPV=(sens·prev)/((sens·prev)+(1-spec)(1-prev)). prev=0.125, sens=0.87, spec=0.78. Compute PPV."}]}, "response": "PPV ≈ 0.361.", "meta": {"id": "WUA_STAT_25K_016998", "domain": "master_scholars_academics", "subdomain": "stats_math::bayes", "task": "quantitative_reasoning", "difficulty": "archon", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "For X~Binomial(n,p), E[X]=np and Var[X]=np(1-p). Given n=35, p=0.22, compute E and Var."}]}, "response": "E[X]=7.70; Var[X]=6.006.", "meta": {"id": "WUA_STAT_25K_016999", "domain": "master_scholars_academics", "subdomain": "stats_math::distributions", "task": "quantitative_reasoning", "difficulty": "core", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics Quantitative Reasoning. Use the formula provided. Show minimal steps and give the final answer with units."}, {"role": "user", "content": "Compute z=(x-μ)/σ. Given x=97.65, μ=17.07, σ=2.24, compute z."}]}, "response": "z = 35.973.", "meta": {"id": "WUA_STAT_25K_017000", "domain": "master_scholars_academics", "subdomain": "stats_math::standardization", "task": "quantitative_reasoning", "difficulty": "advanced", "truth_mode": "self_contained_math", "wrapper": "minimal_chain", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "math"]}, "fact_sources": [{"source_title": "Self-contained in prompt", "source_url": "", "evidence_type": "self_contained", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017001", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017002", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017003", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017004", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017005", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017006", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017007", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017008", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017009", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017010", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017011", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017012", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017013", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017014", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017015", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017016", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017017", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017018", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017019", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017020", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017021", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017022", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017023", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017024", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017025", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017026", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017027", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017028", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017029", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017032", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017033", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017034", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017035", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017036", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017037", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017038", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017039", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017040", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017041", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017042", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017043", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017044", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017045", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017046", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017047", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017048", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017049", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017050", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017051", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017052", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017053", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017054", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017055", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017056", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017057", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017058", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017059", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017060", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017061", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017062", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017063", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017064", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017065", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017066", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017067", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017068", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017069", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017070", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017071", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017072", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017073", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017074", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017075", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017076", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017077", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017078", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017079", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017080", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017081", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017082", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017083", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017084", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017085", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017086", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017087", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017088", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017089", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017090", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017091", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017092", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017093", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017094", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017095", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017096", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017097", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017098", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017099", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017100", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017101", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017103", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017104", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017105", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017106", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017107", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017108", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017109", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017110", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017111", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017112", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017113", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017114", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017115", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017116", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017117", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017118", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017119", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017121", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017122", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017123", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017124", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017125", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017126", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017127", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017128", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017129", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017130", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017131", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017132", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017133", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017134", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017135", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017136", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017137", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017138", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017139", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017140", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017141", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017142", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017143", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017144", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017145", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017146", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017147", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017148", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017149", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017150", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017151", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017152", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017153", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017154", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017155", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017156", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017157", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017158", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017159", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017160", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017161", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017162", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017163", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017164", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017165", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017166", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017167", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017168", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017169", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017170", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017171", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017172", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017173", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017174", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017175", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017176", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017177", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017178", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017179", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017180", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017181", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017182", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017183", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017184", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017185", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017186", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017187", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017188", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017189", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017190", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017191", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017192", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017193", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017194", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017195", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017196", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017197", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017198", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017199", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017200", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017201", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017202", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017203", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017204", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017205", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017206", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017207", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017208", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017209", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017210", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017211", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017212", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017213", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017214", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017215", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017216", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017217", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017218", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017219", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017220", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017221", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017222", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017223", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017224", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017225", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017226", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017227", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017228", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017229", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017230", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017231", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017232", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017233", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017234", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017235", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017236", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017237", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017238", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017239", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017240", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017241", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017242", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017243", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017244", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017245", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017246", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017247", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017248", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017249", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017250", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017251", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017252", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017253", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017254", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017255", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017256", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017257", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017258", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017259", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017260", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017261", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017262", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017263", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017264", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017265", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017266", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017267", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017268", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017269", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017270", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017271", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017272", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017273", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017274", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017275", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017276", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017277", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017278", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017279", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017280", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017281", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017282", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017283", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017284", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017285", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017286", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017287", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017288", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017289", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017290", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017291", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017292", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017293", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017294", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017295", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017297", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017298", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017299", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017302", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017303", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017304", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017305", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017306", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017307", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017308", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017309", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017310", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017311", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017312", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017313", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017314", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017315", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017316", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017317", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017318", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017319", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017320", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017321", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017322", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017323", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017324", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017325", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017326", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017327", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017328", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017329", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017330", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017331", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017332", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017333", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017334", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017335", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017336", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017337", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017338", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017339", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017340", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017341", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017342", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017343", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017344", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017345", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017346", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017347", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017348", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017349", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017350", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017351", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017352", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017353", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017354", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017356", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017357", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017358", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017359", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017360", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017361", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017362", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017363", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017364", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017365", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017366", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017367", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017369", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017370", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017371", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017374", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017375", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017376", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017377", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017378", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017379", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017380", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017381", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017382", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017383", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017384", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017385", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017386", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017387", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017388", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017389", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017390", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017391", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017392", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017393", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017394", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017395", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017396", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017397", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017398", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017399", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017400", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017401", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017402", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017403", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017404", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017405", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017406", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017407", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017408", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017409", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017410", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017411", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017412", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017413", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017414", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017415", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017416", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017417", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017418", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017419", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017420", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017421", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017422", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017423", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017424", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017425", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017426", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017427", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017428", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017429", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017430", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017431", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017432", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017433", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017434", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017435", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017436", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017437", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017438", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017439", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017440", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017441", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017442", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017443", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017444", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017445", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017446", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017447", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017448", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017449", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017450", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017451", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017452", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017453", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017454", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017455", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017456", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017457", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017458", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017459", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017460", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017461", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017462", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017463", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017464", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017465", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017466", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017467", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017468", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017469", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017470", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017471", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017472", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017473", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017474", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017475", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017476", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017477", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017478", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017479", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017480", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017481", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017482", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017483", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017484", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017485", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017486", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017487", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017488", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017489", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017490", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017491", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017492", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017493", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017494", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017495", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017496", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017497", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017498", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017499", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017500", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017501", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017502", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017503", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017504", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017505", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017506", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017507", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017508", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017509", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017510", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017511", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017512", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017513", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017514", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017515", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017516", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017517", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017518", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017519", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017520", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017521", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017522", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017523", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017524", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017525", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017526", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017527", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017528", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017529", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017530", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017531", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017532", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017533", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017534", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017535", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017536", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017537", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017538", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017539", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017540", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017541", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017542", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017543", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017544", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017545", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017546", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017547", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017548", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017549", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017550", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017551", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017552", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017553", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017554", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017555", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017556", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017557", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017558", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017559", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017560", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017561", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017562", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017563", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017564", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017565", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017566", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017567", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017568", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017569", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017570", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017571", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017572", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017573", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017574", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017575", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017576", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017577", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017578", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017579", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017580", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017581", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017582", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017583", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017584", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017585", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017586", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017587", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017588", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017589", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017590", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017591", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017592", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017593", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017594", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017595", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017596", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017597", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017598", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017599", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017600", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017601", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017602", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017603", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017604", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017605", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017606", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017607", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017608", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017609", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017610", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017611", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017612", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017613", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017614", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017615", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017616", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017617", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017618", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017619", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017620", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017621", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017622", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017623", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017624", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017625", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017626", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017627", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017628", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017629", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017630", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017631", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017632", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017633", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017634", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017635", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017636", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017637", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017638", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017639", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017640", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017641", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017642", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017643", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017644", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017645", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017646", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017647", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017648", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017649", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017650", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017651", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017652", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017653", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017654", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017655", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017656", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017657", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017658", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017659", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017660", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017661", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017662", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017663", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017664", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017665", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017666", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017667", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017668", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017669", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017670", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017671", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017672", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017673", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017674", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017675", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017676", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017677", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017678", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017679", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017680", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017681", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017682", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017683", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017684", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017685", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017686", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017687", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017688", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017689", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017690", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017691", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017692", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017693", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017694", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017695", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017696", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017697", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017698", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017699", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017700", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017701", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017702", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017703", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017704", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017705", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017706", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017707", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017708", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017709", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017710", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017711", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017712", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017713", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017714", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017715", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017716", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017717", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017718", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017719", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017720", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017721", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017722", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017723", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017724", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017725", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017726", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017727", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017728", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017729", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017730", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017731", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017732", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017733", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017734", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017735", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017736", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017737", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017738", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017739", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017740", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017741", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017742", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017743", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017744", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017745", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017746", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017747", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017748", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017749", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017750", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017751", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017752", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017753", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017754", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017755", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017756", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017757", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017758", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017759", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017760", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017761", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017762", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017763", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017764", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017765", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017766", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017767", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017768", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017769", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017770", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017771", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017772", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017773", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017774", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017775", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017776", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017777", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017778", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017779", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017780", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017781", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017782", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017783", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017784", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017785", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017786", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017787", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017788", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017789", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017790", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017791", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017792", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017793", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017794", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017795", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017796", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017797", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017798", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017799", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017800", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017801", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017802", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017803", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017804", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017805", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017806", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017807", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017808", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017809", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017810", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017811", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017812", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017813", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017814", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017815", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017816", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017817", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017818", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017819", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017820", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017821", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017822", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017823", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017824", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017825", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017826", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017827", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017828", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017829", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017830", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017831", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017832", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017833", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017834", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017835", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017836", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017837", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017838", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017839", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017840", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017841", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017842", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017843", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017844", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017845", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017846", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017847", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017848", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017849", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017850", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017851", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017852", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017853", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017854", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017855", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017856", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017857", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017858", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017859", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017860", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017861", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017862", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017863", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017864", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017865", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017866", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017867", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017868", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017869", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017870", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017871", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017872", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017873", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017874", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017875", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017876", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017877", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017878", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017879", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017880", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017881", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017882", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017883", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017884", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017885", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017886", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017887", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017888", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017889", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017890", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017891", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017892", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017893", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017894", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017895", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017896", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017897", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017898", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017899", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017900", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017901", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017902", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017903", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017904", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017905", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017906", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017907", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017908", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017909", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017910", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017911", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017912", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017913", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017914", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017915", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017916", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017917", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017918", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017919", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017920", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017921", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017922", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017923", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017924", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017925", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017926", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017928", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017929", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017930", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017932", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017933", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017934", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017935", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017936", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017937", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017938", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017939", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017940", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017941", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017942", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017943", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017944", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017945", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017946", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017947", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017948", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017949", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017950", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017951", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017952", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017953", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017954", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017955", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017956", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017957", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017958", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017959", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017960", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017961", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017962", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017963", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017964", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017965", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017966", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017967", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017968", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017969", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017970", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017971", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017972", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017973", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017974", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017975", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017976", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017977", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017978", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017979", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017980", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017981", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017982", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017983", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017984", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017985", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017986", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017987", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017988", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017989", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017990", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017991", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017992", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017993", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017994", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017995", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_017996", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017997", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_017998", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_017999", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018000", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018001", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018002", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018003", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018004", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018005", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018006", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018007", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018008", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018009", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018010", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018011", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018012", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018013", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018014", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018015", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018016", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018017", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018018", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018019", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018020", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018021", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018022", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018023", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018024", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018025", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018026", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018027", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018028", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018029", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018030", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018031", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018032", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018033", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018034", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018035", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018036", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018037", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018038", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018039", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018040", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018041", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018042", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018043", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018044", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018045", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018046", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018047", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018048", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018049", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018050", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018051", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018052", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018053", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018054", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018055", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018056", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018057", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018058", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018059", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018060", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018061", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018062", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018063", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018064", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018065", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018066", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018067", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018068", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018069", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018070", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018071", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018072", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018073", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018074", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018075", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018076", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018077", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018078", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018079", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018080", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018081", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018082", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018083", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018084", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018085", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018086", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018087", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018088", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018089", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018090", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018091", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018092", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018093", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018094", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018095", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018096", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018097", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018098", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018099", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018100", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018101", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018102", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018103", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018104", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018105", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018106", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018107", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018108", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018109", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018110", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018111", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018112", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018113", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018114", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018115", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018116", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018117", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018118", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018119", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018120", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018121", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018122", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018123", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018124", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018125", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018126", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018127", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018128", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018129", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018130", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018131", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018132", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018133", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018134", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018135", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018136", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018137", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018138", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018139", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018140", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018141", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018142", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018143", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018144", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018145", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018146", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018147", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018148", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018149", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018150", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018151", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018152", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018153", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018154", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018155", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018156", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018157", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018158", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018159", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018160", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018161", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018162", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018163", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018164", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018166", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018167", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018168", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018169", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018170", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018171", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018172", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018173", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018174", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018175", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018176", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018177", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018178", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018179", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018180", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018181", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018182", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018183", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018184", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018185", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018186", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018187", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018188", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018189", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018190", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018191", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018192", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018193", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018194", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018195", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018196", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018197", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018198", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018199", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018200", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018201", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018202", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018203", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018204", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018205", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018206", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018207", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018208", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018209", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018210", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018211", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018212", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018213", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018214", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018215", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018216", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018217", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018218", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018219", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018220", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018221", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018222", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018223", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018224", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018225", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018226", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018227", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018228", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018229", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018230", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018232", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018233", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018238", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018239", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018240", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018241", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018242", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018243", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018244", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018245", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018246", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018247", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018248", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018249", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018250", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018251", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018252", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018253", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018254", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018255", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018256", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018257", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018258", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018259", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018260", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018261", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018262", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018263", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018264", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018265", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018266", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018267", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018268", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018269", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018270", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018271", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018272", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018274", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018275", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018276", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018277", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018278", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018279", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018280", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018281", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018282", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018283", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018284", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018285", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018286", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018287", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018288", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018289", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018290", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018291", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018292", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018293", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018294", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018295", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018296", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018297", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018298", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018299", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018300", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018301", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018302", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018303", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018304", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018305", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018306", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018307", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018308", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018309", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018310", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018311", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018312", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018313", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018314", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018315", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018316", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018317", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018318", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018319", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018320", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018321", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018322", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018323", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018324", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018325", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018326", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018327", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018328", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018329", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018330", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018331", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018332", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018333", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018334", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018335", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018336", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018337", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018338", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018339", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018340", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018341", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018342", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018343", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018344", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018345", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018346", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018347", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018348", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018349", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018350", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018351", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018352", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018353", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018354", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018355", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018356", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018357", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018358", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018359", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018360", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018361", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018363", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018364", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018365", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018366", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018367", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018368", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018369", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018370", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018371", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018372", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018373", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018374", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018375", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018376", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018377", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018378", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018379", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018380", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018381", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018382", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018383", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018384", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018385", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018386", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018387", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018388", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018389", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018390", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018391", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018392", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018393", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018394", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018395", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018396", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018397", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018398", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018399", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018400", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018401", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018402", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018403", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018404", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018405", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018406", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018407", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018408", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018409", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018410", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018411", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018412", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018413", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018414", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018415", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018416", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018417", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018418", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018419", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018420", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018421", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018422", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018423", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018424", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018425", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018426", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018427", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018428", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018429", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018430", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018431", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018432", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018433", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018434", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018435", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018436", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018437", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018438", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018439", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018440", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018441", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018442", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018443", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018444", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018445", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018446", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018447", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018448", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018449", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018450", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018451", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018452", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018453", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018454", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018455", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018456", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018457", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018458", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018459", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018460", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018461", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018462", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018463", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018464", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018465", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018466", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018467", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018468", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018469", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018470", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018471", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018472", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018473", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018474", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018475", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018476", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018477", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018478", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018479", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018480", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018481", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018482", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018483", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018484", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018485", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018486", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018487", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018488", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018489", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018490", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018491", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018492", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018493", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018494", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018495", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018496", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018497", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018498", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018499", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018500", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018501", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018502", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018503", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018504", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018505", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018506", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018507", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018508", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018509", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018510", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018511", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018512", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018513", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018514", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018515", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018516", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018517", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018518", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018519", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018520", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018521", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018522", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018523", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018524", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018525", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018526", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018527", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018528", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018529", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018530", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018531", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018532", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018533", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018534", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018535", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018536", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018537", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018538", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018539", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018540", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018541", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018542", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018543", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018544", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018545", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018546", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018547", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018548", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018549", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018550", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018551", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018552", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018553", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018554", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018555", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018556", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018557", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018558", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018559", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018560", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018561", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018562", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018563", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018564", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018565", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018566", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018567", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018568", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018569", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018570", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018571", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018572", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018573", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018574", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018575", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018576", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018577", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018578", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018579", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018580", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018581", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018582", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018583", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018584", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018585", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018586", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018587", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018588", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018589", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018590", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018591", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018592", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018593", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018594", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018595", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018596", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018597", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018598", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018599", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018600", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018601", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018602", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018603", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018604", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018605", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018606", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018607", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018608", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018609", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018610", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018611", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018612", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018613", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018614", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018615", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018616", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018617", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018618", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018619", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018620", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018621", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018622", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018623", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018624", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018625", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018626", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018627", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018628", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018629", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018630", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018631", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018632", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018633", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018634", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018635", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018636", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018637", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018638", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018639", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018640", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018641", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018642", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018643", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018644", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018645", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018646", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018647", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018648", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018649", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018650", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018651", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018652", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018653", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018654", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018655", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018656", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018657", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018658", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018659", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018660", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018661", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018662", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018663", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018664", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018665", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018666", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018667", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018668", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018669", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018670", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018671", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018672", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018673", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018674", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018675", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018676", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018677", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018678", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018679", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018680", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018681", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018682", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018683", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018684", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018685", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018686", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018687", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018688", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018689", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018690", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018691", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018692", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018693", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018694", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018695", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018696", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018697", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018698", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018699", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018700", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018701", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018702", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018703", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018704", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018705", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018706", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018707", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018708", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018709", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018710", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018711", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018712", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018713", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018714", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018715", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018716", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018717", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018718", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018719", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018720", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018721", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018722", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018723", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018724", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018725", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018726", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018727", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018728", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018729", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018730", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018731", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018732", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018733", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018734", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018735", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018736", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018737", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018738", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018739", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018740", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018741", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018742", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018743", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018744", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018745", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018746", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018747", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018748", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018749", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018750", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018751", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018752", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018753", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018754", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018755", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018756", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018757", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018758", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018759", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018760", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018761", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018762", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018763", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018764", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018765", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018766", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018767", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018768", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018769", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018770", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018771", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018772", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018773", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018774", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018775", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018776", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018777", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018778", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018779", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018780", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018781", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018782", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018783", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018784", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018785", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018786", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018787", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018788", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018789", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018790", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018791", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018792", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018793", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018794", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018795", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018796", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018797", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018798", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018799", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018800", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018801", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018802", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018803", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018804", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018805", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018806", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018807", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018808", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018809", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018810", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018811", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018812", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018813", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018814", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018815", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018816", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018817", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018818", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018819", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018820", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018821", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018822", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018823", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018824", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018825", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018826", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018827", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018828", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018829", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018830", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018831", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018832", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018833", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018834", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018835", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018836", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018837", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018838", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018839", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018840", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018841", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018842", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018843", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018844", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018845", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018846", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018847", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018848", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018849", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018850", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018851", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018852", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018853", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018854", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018855", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018856", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018857", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018858", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018859", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018860", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018861", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018862", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018863", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018864", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018865", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018866", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018867", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018868", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018869", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018870", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018871", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018872", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018873", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018874", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018875", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018876", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018877", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018878", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018879", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018880", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018881", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018882", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018883", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018884", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018885", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018886", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018887", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018888", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018889", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018890", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018891", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018892", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018893", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018894", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018895", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018896", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018897", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018898", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018899", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018900", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018901", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018902", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018903", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018904", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018905", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018906", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018907", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018908", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018909", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018910", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018911", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018912", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018913", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018914", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018915", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018916", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018917", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018918", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018919", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018920", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018921", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018922", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018923", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018924", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018925", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018926", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018927", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018928", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018929", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018930", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018931", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018932", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018933", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018934", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018935", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018936", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018937", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018938", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018939", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018940", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018941", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018942", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018943", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018944", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018945", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018946", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018947", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018948", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018949", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018950", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018951", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018952", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018953", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018954", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018955", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018956", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018957", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018958", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018959", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018960", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018961", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018962", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018963", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018964", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018965", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018966", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018967", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018968", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018969", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018970", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018971", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018972", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018973", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018974", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018975", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018976", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018977", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018978", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018979", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018980", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018981", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018982", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018983", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018984", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018985", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018986", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018987", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018988", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018989", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018990", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018991", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018992", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018993", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018994", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018995", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_018996", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_018997", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018998", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_018999", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019000", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019001", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019002", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019003", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019004", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019005", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019006", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019007", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019008", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019009", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019010", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019011", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019012", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019013", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019014", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019015", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019016", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019017", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019018", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019019", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019020", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019021", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019022", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019023", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019024", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019025", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019026", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019027", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019028", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019029", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019030", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019031", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019032", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019033", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019034", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019035", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019036", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019037", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019038", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019039", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019040", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019041", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019042", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019043", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019044", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019045", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019046", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019047", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019048", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019049", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019050", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019051", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019052", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019053", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019054", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019055", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019056", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019057", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019058", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019059", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019060", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019061", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019062", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019063", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019064", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019065", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019066", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019067", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019068", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019069", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019070", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019071", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019072", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019073", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019074", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019075", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019076", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019077", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019078", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019079", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019080", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019081", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019082", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019083", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019084", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019085", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019086", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019087", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019088", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019089", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019090", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019091", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019092", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019093", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019094", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019095", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019096", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019097", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019098", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019099", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019100", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019101", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019102", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019103", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019104", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019105", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019106", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019107", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019108", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019109", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019110", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019111", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019112", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019113", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019114", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019115", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019116", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019117", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019118", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019119", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019120", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019121", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019122", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019123", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019124", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019125", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019126", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019127", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019128", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019129", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019130", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019131", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019132", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019133", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019134", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019135", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019138", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019140", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019141", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019142", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019143", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019144", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019145", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019146", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019147", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019148", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019149", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019150", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019151", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019152", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019153", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019154", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019155", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019156", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019157", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019158", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019159", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019160", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019161", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019162", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019163", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019164", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019165", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019166", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019167", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019168", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019169", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019170", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019171", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019172", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019173", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019174", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019175", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019176", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019177", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019178", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019179", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019180", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019181", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019182", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019183", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019184", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019185", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019186", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019187", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019188", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019189", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019190", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019191", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019192", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019193", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019194", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019195", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019196", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019197", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019198", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019199", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019200", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019201", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019202", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019203", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019204", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019205", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019206", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019207", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019208", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019209", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019210", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019211", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019212", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019213", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019214", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019215", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019216", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019217", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019218", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019219", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019220", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019221", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019222", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019223", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019224", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019225", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019226", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019227", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019228", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019229", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019230", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019231", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019232", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019233", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019234", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019235", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019236", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019237", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019238", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019239", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019240", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019241", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019242", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019243", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019244", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019245", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019246", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019247", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019248", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019249", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019250", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019251", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019252", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019253", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019254", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019255", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019256", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019257", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019258", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019259", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019260", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019261", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019262", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019263", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019264", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019265", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019266", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019267", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019268", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019269", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019270", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019271", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019272", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019273", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019274", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019275", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019276", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019277", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019278", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019279", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019280", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019281", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019282", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019283", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019284", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019285", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019286", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019287", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019288", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019289", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019290", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019291", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019292", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019293", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019294", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019295", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019296", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019297", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019298", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019299", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019300", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019301", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019302", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019303", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019304", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019305", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019306", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019307", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019308", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019309", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019310", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019311", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019312", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019313", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019314", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019315", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019316", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019317", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019318", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019319", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019320", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019321", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019322", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019323", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019324", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019325", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019326", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019327", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019328", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019329", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019330", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019331", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019332", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019333", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019334", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019335", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019336", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019337", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019338", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019339", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019340", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019341", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019342", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019343", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019344", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019345", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019346", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019347", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019348", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019349", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019350", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019351", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019352", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019353", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019354", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019355", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019356", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019357", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019358", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019359", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019360", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019361", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019362", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019363", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019364", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019365", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019366", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019367", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019368", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019369", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019370", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019371", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019372", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019373", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019374", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019375", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019376", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019377", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019378", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019379", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019380", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019381", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019382", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019383", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019384", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019385", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019386", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019387", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019388", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019390", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019391", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019392", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019393", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019394", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019395", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019396", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019397", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019398", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019399", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019400", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019401", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019402", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019403", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019404", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019405", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019406", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019407", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019408", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019409", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019410", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019411", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019412", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019413", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019414", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019415", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019416", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019417", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019418", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019421", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019422", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019427", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019428", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019429", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019430", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019431", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019432", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019433", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019434", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019435", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019436", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019437", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019438", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019439", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019440", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019441", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019442", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019443", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019444", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019445", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019446", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019447", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019448", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019449", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019450", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019451", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019452", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019453", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019454", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019455", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019456", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019457", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019458", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019459", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019460", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019461", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019462", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019463", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019464", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019465", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019466", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019467", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019468", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019469", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019470", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019471", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019472", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019473", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019474", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019475", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019476", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019477", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019478", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019479", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019480", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019481", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019482", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019483", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019484", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019485", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019486", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019487", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019488", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019489", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019490", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019491", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019492", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019493", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019494", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019495", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019496", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019497", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019498", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019499", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019500", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019501", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019502", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019503", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019504", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019505", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019506", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019507", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019508", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019509", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019510", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019511", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019512", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019513", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019514", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019515", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019516", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019517", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019518", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019519", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019520", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019521", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019522", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019523", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019524", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019525", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019526", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019527", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019528", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019529", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019530", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019531", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019532", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019533", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019534", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019535", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019536", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019537", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019538", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019539", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019540", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019541", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019542", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019543", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019544", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019545", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019546", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019547", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019548", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019549", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019550", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019551", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019552", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019553", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019554", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019555", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019556", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019557", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019558", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019559", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019560", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019561", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019562", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019563", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019564", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019565", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019566", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019567", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019568", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019569", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019570", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019571", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019572", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019573", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019574", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019575", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019576", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019577", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019578", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019579", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019580", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019581", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019582", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019583", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019584", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019585", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019586", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019587", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019588", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019589", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019590", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019591", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019592", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019593", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019594", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019595", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019596", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019597", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019598", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019599", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019600", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019601", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019602", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019603", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019604", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019605", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019606", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019607", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019608", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019609", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019610", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019611", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019612", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019613", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019614", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019615", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019616", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019617", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019618", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019619", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019621", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019622", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019623", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019624", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019625", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019626", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019627", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019628", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019629", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019630", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019631", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019632", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019633", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019634", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019635", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019636", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019637", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019638", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019639", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019640", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019641", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019642", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019643", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019644", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019645", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019646", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019647", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019648", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019649", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019650", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019651", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019652", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019653", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019654", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019655", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019656", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019657", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019658", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019659", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019660", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019661", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019662", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019663", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019664", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019665", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019666", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019667", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019668", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019669", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019670", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019671", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019672", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019673", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019674", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019675", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019676", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019677", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019678", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019679", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019680", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019681", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019682", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019683", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019684", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019685", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019686", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019687", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019688", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019689", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019690", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019691", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019692", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019693", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019694", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019695", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019696", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019697", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019698", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019699", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019700", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019701", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019702", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019703", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019704", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019705", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019706", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019707", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019708", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019709", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019710", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019711", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019712", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019713", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019714", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019715", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019716", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019717", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019718", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019719", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019720", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019721", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019722", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019723", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019724", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019725", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019726", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019727", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019728", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019729", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019730", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019731", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019732", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019733", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019734", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019735", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019736", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019737", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019738", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019739", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019740", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019741", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019742", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019743", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019744", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019745", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019746", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019748", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019750", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019752", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019753", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019754", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019755", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019756", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019757", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019758", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019759", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019760", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019761", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019762", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019763", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019764", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019765", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019766", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019768", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019769", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019770", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019771", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019772", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019773", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019774", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019775", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019776", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019777", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019778", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019779", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019780", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019781", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019782", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019783", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019784", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019785", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019786", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019787", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019788", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019789", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019790", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019791", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019792", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019793", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019794", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019795", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019796", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019797", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019798", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019799", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019800", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019801", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019802", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019803", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019804", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019805", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019806", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019807", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019808", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019809", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019810", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019811", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019812", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019813", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019814", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019815", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019816", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019817", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019818", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019819", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019820", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019821", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019822", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019823", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019824", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019825", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019826", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019827", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019828", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019829", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019830", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019831", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019832", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019833", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019834", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019835", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019836", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019837", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019838", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019839", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019840", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019841", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019842", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019843", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019844", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019845", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019846", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019847", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019848", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019849", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019850", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019851", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019852", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019853", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019854", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019855", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019857", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019858", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019859", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019860", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019861", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019862", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019863", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019864", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019865", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019866", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019867", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019868", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019869", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019870", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019871", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019872", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019873", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019874", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019875", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019876", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019877", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019878", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019879", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019880", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019881", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019882", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019883", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019884", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019885", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019886", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019887", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019888", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019889", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019890", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019891", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019892", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019893", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019894", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019895", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019896", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019897", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019898", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019899", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019900", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019901", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019902", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019903", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019904", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019905", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019906", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019907", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019908", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019909", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019910", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019911", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019912", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019913", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019914", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019915", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019916", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019917", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019918", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019919", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019920", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019921", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019922", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019923", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019924", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019925", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019926", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019927", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019928", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019929", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019930", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019931", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019932", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019933", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019934", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019935", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019936", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019937", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019938", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019939", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019940", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019941", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019942", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019943", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019944", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019945", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019946", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019947", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019948", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019949", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019950", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019951", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019952", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019953", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019954", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019955", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019956", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019957", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019958", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019959", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019960", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019961", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019962", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019963", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019964", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019965", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019966", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019967", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019968", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019969", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019970", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019971", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019972", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019973", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019974", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019975", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019976", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019977", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019978", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019979", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019980", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019981", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019982", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019983", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019984", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019985", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019986", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019987", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019988", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019989", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019990", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019991", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_019992", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019993", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019994", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019995", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_019996", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019997", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019998", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_019999", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020000", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020001", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020002", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020003", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020004", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020005", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020006", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020007", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020008", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020009", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020010", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020011", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020012", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020013", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020014", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020015", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020016", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020017", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020018", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020019", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020020", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020021", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020022", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020023", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020024", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020025", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020026", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020027", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020028", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020029", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020030", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020031", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020032", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020033", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020034", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020035", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020036", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020037", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020038", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020039", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020040", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020041", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020042", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020043", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020044", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020045", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020046", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020047", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020048", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020049", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020050", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020054", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020055", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020057", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020058", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020059", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020060", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020061", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020062", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020063", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020064", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020065", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020066", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020067", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020068", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020069", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020070", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020071", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020072", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020073", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020074", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020075", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020076", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020077", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020078", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020079", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020080", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020081", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020082", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020083", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020084", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020085", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020086", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020087", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020088", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020089", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020090", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020091", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020092", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020093", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020094", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020095", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020096", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020097", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020098", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020099", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020100", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020101", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020102", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020103", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020104", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020105", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020106", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020107", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020108", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020109", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020110", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020111", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020112", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020113", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020114", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020115", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020116", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020117", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020118", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020119", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020120", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020121", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020122", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020123", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020124", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020125", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020126", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020129", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020130", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020131", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020132", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020133", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020134", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020135", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020136", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020137", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020138", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020139", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020140", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020141", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020142", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020143", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020144", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020145", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020146", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020147", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020148", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020149", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020150", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020151", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020152", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020153", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020154", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020155", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020156", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020157", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020158", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020159", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020160", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020161", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020162", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020163", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020164", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020165", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020166", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020167", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020168", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020169", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020170", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020171", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020172", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020173", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020174", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020175", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020176", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020177", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020178", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020179", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020180", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020181", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020182", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020183", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020184", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020185", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020186", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020187", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020188", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020189", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020190", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020191", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020192", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020193", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020194", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020195", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020196", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020197", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020198", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020199", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020200", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020201", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020202", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020203", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020204", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020205", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020206", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020207", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020208", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020209", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020210", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020211", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020212", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020213", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020214", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020215", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020216", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020217", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020218", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020219", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020220", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020221", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020222", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020223", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020224", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020225", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020226", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020227", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020228", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020229", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020230", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020231", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020232", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020233", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020234", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020235", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020236", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020237", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020238", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020239", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020240", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020241", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020242", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020243", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020244", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020245", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020246", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020247", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020248", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020249", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020250", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020251", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020252", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020253", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020254", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020255", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020256", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020257", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020258", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020259", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020260", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020261", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020262", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020263", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020264", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020265", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020266", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020267", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020268", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020269", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020270", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020271", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020272", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020273", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020274", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020275", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020276", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020277", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020278", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020279", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020280", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020281", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020282", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020283", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020284", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020285", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020286", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020287", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020288", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020289", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020290", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020291", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020292", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020293", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020294", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020295", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020296", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020297", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020298", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020299", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020300", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020301", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020302", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020303", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020304", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020305", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020306", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020307", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020308", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020309", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020310", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020311", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020312", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020313", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020314", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020315", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020316", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020317", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020318", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020319", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020320", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020321", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020322", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020323", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020324", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020325", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020326", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020327", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020328", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020329", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020330", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020331", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020332", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020333", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020334", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020335", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020336", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020337", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020338", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020339", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020340", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020341", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020342", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020344", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020345", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020346", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020347", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020348", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020349", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020350", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020351", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020353", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020355", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020360", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020362", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020363", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020364", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020365", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020366", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020367", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020368", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020369", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020370", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020371", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020372", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020373", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020374", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020375", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020376", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020377", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020378", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020379", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020380", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020381", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020382", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020383", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020384", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020385", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020386", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020387", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020388", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020389", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020391", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020392", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020393", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020394", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020397", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020398", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020400", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020401", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020402", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020403", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020404", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020405", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020407", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020409", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020417", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020418", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020419", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020420", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020421", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020422", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020423", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020424", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020425", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020426", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020427", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020428", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020429", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020430", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020431", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020432", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020433", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020434", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020435", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020436", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020437", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020438", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020439", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020440", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020441", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020442", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020443", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020444", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020445", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020446", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020447", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020448", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020449", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020450", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020451", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020452", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020453", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020454", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020455", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020456", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020457", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020458", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020459", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020460", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020461", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020462", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020463", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020464", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020465", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020466", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020467", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020468", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020469", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020470", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020471", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020472", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020473", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020474", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020475", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020476", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020477", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020478", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020479", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020480", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020481", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020482", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020483", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020484", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020485", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020486", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020487", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020488", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020489", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020490", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020491", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020492", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020493", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020494", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020495", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020496", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020497", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020498", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020499", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020500", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020501", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020502", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020503", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020504", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020505", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020506", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020507", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020508", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020509", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020510", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020511", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020512", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020513", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020514", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020515", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020516", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020517", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020518", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020519", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020520", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020521", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020522", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020523", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020524", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020525", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020526", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020527", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020528", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020529", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020530", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020531", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020532", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020533", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020534", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020535", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020536", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020537", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020538", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020539", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020540", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020541", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020542", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020543", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020544", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020545", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020546", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020547", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020548", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020549", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020550", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020551", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020552", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020553", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020554", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020555", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020556", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020557", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020558", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020559", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020560", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020561", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020562", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020563", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020564", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020565", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020566", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020567", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020568", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020569", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020570", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020571", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020572", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020573", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020574", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020575", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020576", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020577", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020578", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020579", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020580", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020581", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020582", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020583", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020584", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020585", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020586", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020587", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020588", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020589", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020590", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020591", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020592", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020593", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020594", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020595", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020596", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020597", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020598", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020599", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020600", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020601", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020602", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020603", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020604", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020605", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020606", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020607", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020608", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020609", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020610", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020611", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020612", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020613", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020614", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020615", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020616", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020617", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020618", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020619", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020620", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020621", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020622", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020623", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020624", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020625", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020626", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020627", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020628", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020629", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020630", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020631", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020632", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020633", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020634", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020635", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020636", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020637", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020638", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020639", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020640", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020641", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020642", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020643", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020644", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020645", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020646", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020647", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020648", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020649", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020650", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020651", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020652", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020653", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020654", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020655", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020656", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020657", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020658", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020659", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020660", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020661", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020662", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020663", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020664", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020665", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020666", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020667", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020668", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020669", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020670", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020671", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020672", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020673", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020674", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020675", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020676", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020677", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020678", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020679", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020680", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020681", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020682", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020683", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020684", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020685", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020686", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020687", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020688", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020689", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020690", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020691", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020692", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020693", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020694", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020695", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020696", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020697", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020698", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020699", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020700", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020701", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020702", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020703", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020704", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020705", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020706", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020707", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020708", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020709", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020710", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020711", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020712", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020713", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020714", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020715", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020716", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020717", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020718", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020719", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020720", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020722", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020724", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020725", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020726", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020727", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020728", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020729", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020730", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020731", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020732", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020733", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020734", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020735", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020736", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020738", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020740", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020741", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020742", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020743", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020744", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020745", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020746", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020747", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020748", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020749", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020750", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020751", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020752", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020753", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020754", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020755", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020756", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020757", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020758", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020759", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020760", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020761", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020762", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020763", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020764", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020765", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020766", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020767", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020768", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020769", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020770", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020771", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020772", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020773", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020774", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020775", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020776", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020777", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020778", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020779", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020780", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020781", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020782", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020783", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020784", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020785", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020786", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020787", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020788", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020789", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020790", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020791", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020792", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020793", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020794", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020795", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020796", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020797", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020798", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020799", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020800", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020801", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020802", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020803", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020804", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020805", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020806", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020807", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020808", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020809", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020810", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020811", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020812", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020813", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020814", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020815", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020816", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020817", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020818", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020819", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020820", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020821", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020822", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020823", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020824", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020825", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020826", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020827", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020828", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020829", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020830", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020831", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020832", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020833", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020834", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020835", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020836", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020837", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020838", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020839", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020840", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020841", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020842", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020843", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020844", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020845", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020847", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020848", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020849", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020850", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020851", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020852", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020853", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020854", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020855", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020856", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020857", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020858", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020859", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020860", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020861", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020862", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020863", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020864", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020865", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020866", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020867", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020868", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020869", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020870", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020871", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020872", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020873", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020874", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020875", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020876", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020877", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020878", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020879", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020880", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020881", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020882", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020883", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020884", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020885", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020886", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020887", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020888", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020889", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020890", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020891", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020892", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020893", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020894", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020895", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020896", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020897", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020898", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020899", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020900", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020901", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020902", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020903", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020904", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020905", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020906", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020907", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020908", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020909", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020910", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020911", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020912", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020913", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020914", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020915", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020916", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020917", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020918", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020919", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020920", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020921", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020922", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020923", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020924", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020925", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020926", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020927", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020928", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020929", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020930", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020931", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020932", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020933", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020934", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020935", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020936", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020937", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020938", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020939", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020940", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020941", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020942", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020943", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020944", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020945", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020946", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020947", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020948", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020949", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020950", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020951", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020952", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020953", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020954", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020955", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020956", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020957", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020958", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020959", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020960", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020961", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020962", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020963", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020964", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020965", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020966", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020967", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020968", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020969", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020970", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020971", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020972", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020973", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020974", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020975", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020976", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020977", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020978", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020979", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020980", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020981", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020982", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020983", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020984", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020985", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020986", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020987", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020988", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020989", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020990", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020991", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020992", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020993", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020994", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_020995", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020996", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_020997", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020998", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_020999", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021000", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021001", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021002", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021003", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021004", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021005", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021006", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021007", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021008", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021009", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021010", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021011", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021012", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021013", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021014", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021015", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021016", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021017", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021018", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021019", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021020", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021021", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021022", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021023", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021024", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021025", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021026", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021027", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021028", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021029", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021030", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021031", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021032", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021033", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021034", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021035", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021036", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021037", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021038", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021039", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021040", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021041", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021042", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021043", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021044", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021045", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021046", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021047", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021048", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021049", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021050", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021051", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021052", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021053", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021054", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021055", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021056", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021057", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021058", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021059", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021060", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021061", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021062", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021063", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021064", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021065", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021066", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021067", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021068", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021069", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021070", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021071", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021072", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021073", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021074", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021075", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021076", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021077", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021078", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021079", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021080", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021081", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021082", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021083", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021084", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021085", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021086", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021087", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021088", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021089", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021090", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021091", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021092", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021093", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021094", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021095", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021096", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021097", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021098", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021099", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021100", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021101", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021102", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021103", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021104", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021105", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021106", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021107", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021108", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021109", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021110", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021111", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021112", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021113", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021114", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021115", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021116", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021117", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021118", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021119", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021120", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021121", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021122", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021123", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021124", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021125", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021126", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021127", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021128", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021129", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021130", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021131", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021132", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021133", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021134", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021135", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021136", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021137", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021138", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021139", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021140", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021141", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021142", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021143", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021144", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021145", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021146", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021147", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021148", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021149", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021150", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021151", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021154", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021155", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021156", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021157", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021158", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021159", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021160", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021161", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021162", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021163", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021164", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021165", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021166", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021167", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021168", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021169", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021170", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021171", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021172", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021173", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021174", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021175", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021176", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021177", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021178", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021179", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021180", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021181", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021182", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021183", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021184", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021185", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021186", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021187", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021188", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021189", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021190", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021191", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021192", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021193", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021194", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021195", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021196", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021197", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021198", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021199", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021200", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021201", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021202", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021203", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021204", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021205", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021206", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021207", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021208", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021209", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021210", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021211", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021212", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021213", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021214", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021215", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021216", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021217", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021218", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021219", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021220", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021221", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021222", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021223", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021224", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021225", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021226", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021227", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021228", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021229", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021230", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021231", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021232", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021233", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021234", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021235", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021236", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021237", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021238", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021239", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021240", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021241", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021242", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021243", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021244", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021245", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021246", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021247", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021248", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021249", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021250", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021251", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021252", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021253", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021254", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021255", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021256", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021257", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021258", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021259", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021260", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021261", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021262", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021263", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021264", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021265", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021266", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021267", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021268", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021269", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021270", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021271", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021272", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021273", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021274", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021275", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021276", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021277", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021278", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021279", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021280", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021281", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021282", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021283", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021284", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021285", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021286", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021287", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021288", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021289", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021290", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021291", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021292", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021293", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021294", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021295", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021296", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021297", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021298", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021299", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021300", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021301", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021302", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021303", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021304", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021305", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021306", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021307", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021308", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021309", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021310", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021311", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021312", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021313", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021314", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021315", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021316", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021317", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021318", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021319", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021320", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021321", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021322", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021323", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021324", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021325", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021326", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021327", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021328", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021329", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021330", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021331", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021332", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021333", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021334", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021335", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021336", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021337", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021338", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021339", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021340", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021341", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021342", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021343", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021344", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021345", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021346", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021347", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021348", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021349", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021350", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021351", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021352", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021353", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021354", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021355", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021356", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021357", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021358", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021359", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021360", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021361", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021362", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021363", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021364", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021365", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021366", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021367", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021368", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021369", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021370", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021371", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021372", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021373", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021374", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021375", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021376", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021377", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021378", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021379", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021380", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021381", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021382", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021383", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021384", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021385", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021386", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021387", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021388", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021389", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021390", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021391", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021392", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021393", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021394", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021395", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021396", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021397", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021398", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021399", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021400", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021401", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021402", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021403", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021404", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021405", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021406", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021407", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021408", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021409", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021410", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021411", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021412", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021413", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021414", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021415", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021416", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021417", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021418", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021419", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021420", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021421", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021422", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021423", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021424", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021425", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021426", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021427", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021428", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021429", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021430", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021431", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021432", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021433", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021434", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021435", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021436", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021437", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021438", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021439", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021440", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021441", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021442", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021443", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021444", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021445", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021446", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021447", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021448", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021449", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021450", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021451", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021452", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021453", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021454", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021455", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021456", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021457", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021458", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021459", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021460", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021461", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021462", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021463", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021464", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021465", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021466", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021467", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021468", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021469", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021470", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021471", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021472", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021473", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021474", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021475", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021476", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021477", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021478", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021479", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021480", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021481", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021482", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021483", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021484", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021485", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021486", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021487", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021488", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021489", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021496", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021498", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021499", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021500", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021501", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021502", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021503", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021504", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021505", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021506", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021507", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021508", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021509", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021510", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021511", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021512", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021513", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021514", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021515", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021516", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021517", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021518", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021519", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021520", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021521", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021522", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021523", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021524", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021525", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021526", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021527", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021528", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021529", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021531", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021532", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021533", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021534", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021535", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021536", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021537", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021538", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021539", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021540", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021541", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021542", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021543", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021544", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021545", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021546", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021547", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021548", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021549", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021550", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021551", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021552", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021553", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021554", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021555", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021556", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021557", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021558", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021559", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021560", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021568", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021569", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021570", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021571", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021572", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021573", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021574", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021575", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021576", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021577", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021578", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021579", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021580", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021581", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021582", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021585", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021586", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021587", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021588", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021589", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021590", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021591", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021592", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021593", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021594", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021595", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021596", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021597", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021598", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021599", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021600", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021601", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021602", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021603", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021604", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021605", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021606", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021607", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021608", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021609", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021610", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021611", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021612", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021613", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021614", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021615", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021616", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021617", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021618", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021619", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021620", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021621", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021622", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021623", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021624", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021625", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021626", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021627", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021628", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021629", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021630", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021631", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021632", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021633", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021634", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021635", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021636", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021637", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021638", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021639", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021640", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021641", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021642", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021643", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021644", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021645", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021646", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021647", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021648", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021649", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021650", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021651", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021652", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021653", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021654", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021655", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021656", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021657", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021658", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021659", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021660", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021661", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021662", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021663", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021664", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021665", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021666", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021667", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021668", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021669", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021670", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021671", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021672", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021673", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021674", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021675", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021676", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021677", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021678", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021679", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021680", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021681", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021682", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021683", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021684", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021685", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021686", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021687", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021688", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021689", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021690", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021691", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021692", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021693", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021694", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021695", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021696", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021697", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021698", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021699", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021700", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021701", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021702", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021703", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021704", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021705", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021706", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021707", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021708", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021709", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021710", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021711", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021712", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021713", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021714", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021715", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021716", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021717", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021718", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021719", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021720", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021721", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021722", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021723", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021724", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021725", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021726", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021727", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021728", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021729", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021730", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021731", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021732", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021733", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021734", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021735", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021736", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021737", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021738", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021739", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021740", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021741", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021742", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021743", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021744", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021745", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021746", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021747", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021748", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021749", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021750", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021751", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021752", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021753", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021754", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021755", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021756", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021757", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021758", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021759", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021760", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021761", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021762", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021763", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021764", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021765", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021766", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021767", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021768", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021769", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021770", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021771", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021772", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021773", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021774", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021775", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021776", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021777", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021778", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021779", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021780", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021781", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021782", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021783", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021784", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021785", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021786", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021787", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021788", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021789", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021790", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021791", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021792", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021793", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021794", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021795", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021796", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021797", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021798", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021799", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021800", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021801", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021802", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021803", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021804", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021805", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021806", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021807", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021808", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021809", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021810", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021811", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021812", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021813", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021814", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021815", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021816", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021818", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021819", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021820", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021821", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021822", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021823", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021824", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021825", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021826", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021827", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021828", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021829", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021830", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021831", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021832", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021833", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021834", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021835", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021836", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021837", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021838", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021839", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021840", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021841", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021842", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021843", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021844", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021845", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021846", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021847", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021848", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021849", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021850", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021851", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021852", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021853", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021854", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021855", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021856", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021857", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021858", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021859", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021860", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021861", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021862", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021863", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021864", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021865", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021866", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021867", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021868", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021869", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021870", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021871", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021872", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021873", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021874", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021875", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021876", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021877", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021878", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021879", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021880", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021881", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021882", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021883", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021884", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021885", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021886", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021887", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021888", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021890", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021891", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021892", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021893", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021894", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021895", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021896", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021897", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021898", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021899", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021900", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021901", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021902", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021903", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021904", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021905", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021906", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021907", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021908", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021909", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021910", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021911", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021912", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021913", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021914", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021915", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021916", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021917", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021918", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021919", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021920", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021921", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021922", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021923", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021924", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021925", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021926", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021927", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021928", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021929", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021930", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021931", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021932", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021933", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021934", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021935", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021936", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021937", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021938", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021939", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021940", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021941", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021942", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021943", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021944", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021945", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021946", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021947", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021948", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021949", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021950", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021951", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021952", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021953", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021954", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021955", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021956", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021957", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021958", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021959", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021960", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021961", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021962", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021963", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021964", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021965", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021966", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021967", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021968", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021969", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021970", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021971", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021972", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021973", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021974", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021975", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021976", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021977", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021978", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021979", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021980", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021981", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021982", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021983", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021984", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021985", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021986", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021987", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021988", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021989", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021990", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021991", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021992", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021993", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021994", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_021995", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021996", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021997", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_021998", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_021999", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022000", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022001", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022002", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022003", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022004", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022005", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022006", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022007", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022008", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022009", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022010", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022011", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022012", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022013", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022014", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022015", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022016", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022017", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022018", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022019", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022020", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022021", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022022", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022023", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022024", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022025", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022026", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022027", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022028", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022029", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022030", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022031", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022032", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022033", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022034", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022035", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022036", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022037", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022038", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022039", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022040", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022041", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022042", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022043", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022044", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022045", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022046", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022047", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022048", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022049", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022050", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022051", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022052", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022053", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022054", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022055", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022056", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022057", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022058", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022059", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022060", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022061", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022062", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022063", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022064", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022065", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022066", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022067", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022068", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022069", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022070", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022071", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022072", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022073", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022074", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022075", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022076", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022077", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022078", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022079", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022080", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022081", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022082", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022083", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022084", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022085", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022086", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022087", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022088", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022089", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022090", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022091", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022092", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022093", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022094", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022095", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022096", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022097", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022098", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022099", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022100", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022101", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022102", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022103", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022104", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022105", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022106", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022107", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022108", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022109", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022110", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022111", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022112", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022113", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022114", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022115", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022116", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022117", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022118", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022119", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022120", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022121", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022122", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022123", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022124", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022125", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022126", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022127", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022128", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022129", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022130", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022131", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022132", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022133", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022134", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022135", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022136", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022137", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022138", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022139", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022140", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022141", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022142", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022143", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022144", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022145", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022146", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022147", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022148", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022149", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022150", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022151", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022152", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022153", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022154", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022155", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022156", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022157", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022158", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022159", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022160", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022161", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022162", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022163", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022164", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022165", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022166", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022167", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022168", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022169", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022170", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022171", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022172", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022173", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022174", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022175", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022176", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022177", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022178", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022179", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022180", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022181", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022182", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022183", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022184", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022185", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022186", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022187", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022188", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022189", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022190", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022195", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022198", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022199", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022200", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022201", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022202", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022203", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022204", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022205", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022206", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022207", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022208", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022209", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022210", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022211", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022212", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022213", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022214", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022215", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022216", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022217", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022218", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022219", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022220", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022221", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022222", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022223", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022224", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022225", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022226", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022227", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022228", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022229", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022230", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022231", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022234", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022235", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022236", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022237", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022238", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022239", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022240", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022241", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022242", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022243", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022244", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022245", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022246", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022247", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022248", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022249", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022250", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022251", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022252", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022253", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022254", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022255", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022256", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022257", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022258", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022259", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022260", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022261", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022262", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022263", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022264", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022265", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022266", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022267", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022268", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022269", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022270", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022271", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022272", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022273", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022274", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022275", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022276", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022277", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022278", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022279", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022280", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022281", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022282", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022283", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022284", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022285", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022286", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022287", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022288", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022289", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022290", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022291", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022292", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022293", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022294", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022295", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022296", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022297", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022298", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022299", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022300", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022301", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022302", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022303", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022306", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022307", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022308", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022309", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022310", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022311", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022312", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022313", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022314", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022315", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022316", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022317", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022318", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022319", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022320", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022321", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022323", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022324", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022325", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022326", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022327", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022328", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022329", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022330", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022331", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022332", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022333", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022334", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022335", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022336", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022337", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022338", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022339", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022341", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022344", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022345", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022347", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022348", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022349", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022350", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022351", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022353", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022354", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022355", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022357", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022359", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022360", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022361", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022362", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022363", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022364", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022365", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022366", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022367", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022368", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022369", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022370", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022371", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022372", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022373", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022374", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022375", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022376", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022377", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022379", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022380", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022381", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022382", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022383", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022384", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022385", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022386", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022387", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022388", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022389", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022390", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022391", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022392", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022393", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022396", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022397", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022398", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022399", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022400", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022401", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022402", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022403", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022404", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022405", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022406", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022407", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022408", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022409", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022410", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022411", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022412", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022413", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022414", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022415", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022416", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022417", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022418", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022419", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022420", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022421", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022422", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022423", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022424", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022425", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022426", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022427", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022428", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022429", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022430", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022431", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022432", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022433", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022434", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022435", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022436", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022437", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022438", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022439", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022440", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022441", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022442", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022443", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022444", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022445", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022446", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022447", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022448", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022449", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022450", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022451", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022452", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022453", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022454", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022455", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022456", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022457", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022458", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022459", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022460", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022461", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022462", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022463", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022464", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022465", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022466", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022467", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022468", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022469", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022470", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022471", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022472", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022473", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022474", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022475", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022476", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022477", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022478", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022479", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022480", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022481", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022482", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022483", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022484", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022485", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022486", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022487", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022488", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022489", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022490", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022491", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022492", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022493", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022494", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022495", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022496", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022497", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022498", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022499", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022500", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022501", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022502", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022503", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022504", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022505", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022506", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022507", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022508", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022509", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022510", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022511", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022512", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022513", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022514", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022515", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022516", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022517", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022518", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022519", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022520", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022521", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022522", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022523", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022524", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022525", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022526", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022527", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022528", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022529", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022530", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022531", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022532", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022533", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022534", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022535", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022536", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022537", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022538", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022539", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022540", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022541", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022542", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022543", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022544", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022545", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022546", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022547", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022548", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022549", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022550", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022551", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022552", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022553", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022554", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022555", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022556", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022558", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022559", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022560", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022561", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022562", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022563", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022564", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022565", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022566", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022567", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022568", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022569", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022570", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022571", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022572", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022573", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022574", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022575", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022576", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022577", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022578", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022579", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022580", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022581", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022582", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022583", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022584", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022585", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022586", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022587", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022588", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022589", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022590", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022591", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022592", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022593", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022594", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022595", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022596", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022597", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022598", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022599", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022600", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022601", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022602", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022603", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022604", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022605", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022606", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022607", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022608", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022609", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022610", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022611", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022612", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022613", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022614", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022615", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022616", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022617", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022618", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022619", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022620", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022621", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022622", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022623", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022624", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022625", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022626", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022627", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022628", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022629", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022630", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022631", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022632", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022633", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022634", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022635", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022636", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022637", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022638", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022639", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022640", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022641", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022642", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022643", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022644", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022645", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022646", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022648", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022649", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022650", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022651", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022652", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022653", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022654", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022655", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022656", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022657", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022658", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022659", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022661", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022662", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022668", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022669", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022670", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022671", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022672", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022673", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022674", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022676", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022677", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022678", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022679", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022680", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022684", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022685", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022686", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022687", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022688", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022689", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022690", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022691", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022692", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022693", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022694", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022695", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022696", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022697", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022698", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022699", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022702", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022703", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022704", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022705", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022706", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022707", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022708", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022709", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022710", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022711", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022712", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022713", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022714", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022715", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022716", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022717", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022718", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022719", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022720", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022721", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022722", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022723", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022724", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022725", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022726", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022727", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022728", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022729", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022730", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022731", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022732", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022733", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022734", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022735", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022736", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022737", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022738", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022739", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022740", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022741", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022742", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022743", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022744", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022745", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022746", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022747", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022748", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022749", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022750", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022751", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022752", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022753", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022754", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022755", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022756", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022757", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022758", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022759", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022760", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022761", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022762", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022763", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022764", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022765", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022766", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022767", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022768", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022769", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022770", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022771", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022772", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022773", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022774", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022775", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022776", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022777", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022778", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022779", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022780", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022781", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022782", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022783", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022784", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022785", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022786", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022787", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022788", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022789", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022790", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022791", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022792", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022793", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022794", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022795", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022796", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022797", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022798", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022799", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022800", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022801", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022802", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022803", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022804", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022805", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022806", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022807", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022808", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022809", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022811", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022812", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022813", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022814", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022815", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022816", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022817", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022818", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022819", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022820", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022821", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022822", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022823", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022824", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022828", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022829", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022830", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022831", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022832", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022833", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022834", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022835", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022836", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022837", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022838", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022839", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022840", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022841", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022842", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022843", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022844", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022845", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022846", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022847", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022848", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022849", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022850", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022851", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022852", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022853", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022854", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022855", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022856", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022857", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022858", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022859", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022860", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022861", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022862", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022863", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022864", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022865", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022866", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022867", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022868", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022869", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022870", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022871", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022872", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022873", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022874", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022875", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022876", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022877", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022878", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022879", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022880", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022881", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022882", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022883", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022884", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022885", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022886", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022887", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022888", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022889", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022890", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022891", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022892", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022893", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022894", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022895", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022896", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022897", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022898", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022899", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022900", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022901", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022902", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022903", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022904", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022905", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022906", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022907", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022908", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022909", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022910", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022911", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022912", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022913", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022914", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022915", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022916", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022917", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022918", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022919", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022920", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022921", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022922", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022923", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022924", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022925", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022926", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022927", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022928", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022929", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022930", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022931", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022932", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022933", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022934", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022935", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022936", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022937", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022938", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022939", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022940", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022941", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022942", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022943", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022944", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022945", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022946", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022947", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022948", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022949", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022950", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022951", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022952", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022953", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022954", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022955", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022956", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022957", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022958", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022959", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022960", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022961", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022962", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022963", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022964", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022965", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022966", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022967", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022968", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022969", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022970", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022971", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022972", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022973", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022974", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022975", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022976", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022977", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022978", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022979", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022980", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022981", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022982", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022983", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022984", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022985", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022986", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022987", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022988", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022989", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022990", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022991", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022992", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022993", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022994", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_022995", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022996", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_022997", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022998", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_022999", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023000", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023001", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023002", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023003", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023004", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023005", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023006", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023007", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023008", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023009", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023010", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023011", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023012", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023013", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023014", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023015", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023016", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023017", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023018", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023019", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023020", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023021", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023022", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023023", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023024", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023025", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023026", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023027", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023028", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023029", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023030", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023031", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023032", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023033", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023034", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023035", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023036", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023037", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023038", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023039", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023040", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023041", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023042", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023043", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023044", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023045", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023046", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023047", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023048", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023049", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023050", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023051", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023052", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023053", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023054", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023055", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023056", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023057", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023058", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023059", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023060", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023061", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023062", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023063", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023064", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023065", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023066", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023067", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023068", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023069", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023070", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023071", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023072", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023073", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023074", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023075", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023076", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023077", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023078", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023079", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023080", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023081", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023082", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023083", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023084", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023085", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023086", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023087", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023088", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023089", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023090", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023091", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023092", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023093", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023094", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023095", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023096", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023097", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023098", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023099", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023100", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023101", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023102", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023103", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023104", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023105", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023106", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023107", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023108", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023109", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023111", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023112", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023113", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023115", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023116", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023117", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023118", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023119", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023120", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023121", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023122", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023123", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023124", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023125", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023126", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023127", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023128", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023129", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023130", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023131", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023132", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023133", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023134", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023135", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023136", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023137", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023138", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023139", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023140", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023141", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023142", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023143", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023144", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023145", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023146", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023147", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023148", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023149", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023150", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023151", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023152", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023153", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023154", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023155", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023156", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023157", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023158", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023159", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023160", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023161", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023162", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023163", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023164", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023165", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023166", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023167", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023168", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023169", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023170", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023171", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023172", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023173", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023174", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023175", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023176", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023177", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023178", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023179", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023180", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023181", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023182", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023183", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023184", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023185", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023186", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023187", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023188", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023189", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023190", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023191", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023192", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023193", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023194", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023195", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023196", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023197", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023198", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023199", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023200", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023201", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023202", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023203", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023204", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023205", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023206", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023207", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023208", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023209", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023210", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023211", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023212", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023213", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023214", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023215", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023216", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023217", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023218", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023219", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023220", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023221", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023222", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023223", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023224", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023225", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023226", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023227", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023228", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023229", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023230", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023231", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023232", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023233", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023234", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023235", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023236", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023237", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023238", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023239", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023240", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023241", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023242", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023243", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023244", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023245", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023246", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023247", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023248", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023249", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023250", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023251", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023252", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023253", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023254", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023255", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023256", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023257", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023258", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023259", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023260", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023261", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023262", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023263", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023264", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023265", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023266", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023267", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023268", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023269", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023270", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023271", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023272", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023273", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023274", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023275", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023276", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023277", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023278", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023279", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023280", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023281", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023282", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023283", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023284", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023285", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023286", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023287", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023288", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023289", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023290", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023291", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023292", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023293", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023294", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023295", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023296", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023297", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023298", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023299", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023300", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023301", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023302", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023303", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023304", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023305", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023306", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023307", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023308", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023309", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023310", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023311", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023312", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023313", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023314", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023315", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023316", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023317", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023318", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023319", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023320", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023321", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023322", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023324", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023326", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023327", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023328", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023329", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023330", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023332", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023333", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023334", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023335", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023336", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023337", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023338", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023339", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023340", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023341", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023342", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023343", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023344", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023345", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023346", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023347", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023348", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023349", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023350", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023351", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023352", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023353", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023354", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023355", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023356", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023357", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023358", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023359", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023360", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023361", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023362", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023363", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023364", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023365", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023366", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023368", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023369", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023370", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023371", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023372", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023373", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023374", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023375", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023376", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023377", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023378", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023379", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023380", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023381", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023382", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023383", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023384", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023385", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023386", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023387", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023388", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023389", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023390", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023391", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023392", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023393", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023394", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023395", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023396", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023397", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023398", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023399", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023400", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023401", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023402", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023403", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023404", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023405", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023406", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023407", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023408", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023409", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023410", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023411", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023412", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023413", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023414", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023415", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023416", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023417", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023418", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023419", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023420", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023421", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023422", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023423", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023424", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023425", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023426", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023427", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023428", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023429", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023430", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023431", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023432", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023433", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023434", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023435", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023436", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023437", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023438", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023439", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023440", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023441", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023442", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023443", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023444", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023445", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023446", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023447", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023448", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023449", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023450", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023451", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023452", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023453", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023454", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023455", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023456", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023457", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023458", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023459", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023460", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023461", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023462", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023463", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023464", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023465", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023466", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023467", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023468", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023469", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023470", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023471", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023472", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023473", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023474", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023475", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023476", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023477", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023478", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023479", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023480", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023481", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023482", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023483", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023484", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023485", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023486", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023487", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023488", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023489", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023490", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023491", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023492", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023493", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023494", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023495", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023496", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023497", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023498", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023499", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023500", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023501", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023502", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023503", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023504", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023505", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023506", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023507", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023508", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023509", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023510", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023511", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023512", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023513", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023514", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023515", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023516", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023517", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023518", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023519", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023520", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023521", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023522", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023523", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023524", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023525", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023526", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023527", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023528", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023529", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023530", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023531", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023532", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023533", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023534", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023535", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023536", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023537", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023538", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023539", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023540", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023541", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023542", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023543", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023544", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023545", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023546", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023547", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023548", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023549", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023550", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023551", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023552", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023553", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023554", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023555", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023556", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023557", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023558", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023559", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023560", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023561", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023562", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023563", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023564", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023565", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023566", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023567", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023568", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023569", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023570", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023571", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023572", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023573", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023574", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023575", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023576", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023577", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023578", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023579", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023580", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023581", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023582", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023583", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023584", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023585", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023586", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023587", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023588", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023589", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023590", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023591", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023592", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023593", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023594", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023595", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023596", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023597", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023598", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023599", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023600", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023601", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023602", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023603", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023604", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023605", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023606", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023607", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023608", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023609", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023610", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023611", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023612", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023613", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023614", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023615", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023616", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023617", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023618", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023619", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023620", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023621", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023622", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023623", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023624", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023625", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023626", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023627", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023628", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023629", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023630", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023631", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023632", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023633", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023634", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023635", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023636", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023637", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023638", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023639", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023640", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023641", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023642", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023643", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023644", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023645", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023646", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023647", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023648", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023649", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023650", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023651", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023652", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023653", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023656", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023657", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023658", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023659", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023660", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023661", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023662", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023663", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023664", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023665", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023666", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023667", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023668", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023669", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023670", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023671", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023672", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023673", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023674", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023675", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023676", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023677", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023678", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023679", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023680", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023681", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023682", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023683", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023684", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023685", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023686", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023687", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023688", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023689", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023690", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023691", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023692", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023693", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023694", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023695", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023696", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023697", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023698", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023699", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023700", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023701", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023702", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023703", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023704", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023705", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023706", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023707", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023708", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023709", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023710", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023711", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023712", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023713", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023714", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023715", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023716", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023717", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023718", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023719", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023720", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023721", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023722", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023723", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023724", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023725", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023726", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023727", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023728", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023729", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023730", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023731", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023732", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023733", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023734", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023735", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023736", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023737", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023738", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023739", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023740", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023741", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023742", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023743", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023744", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023745", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023746", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023747", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023748", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023749", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023750", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023751", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023752", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023753", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023754", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023755", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023756", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023757", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023758", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023759", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023760", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023761", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023763", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023764", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023765", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023766", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023767", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023768", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023769", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023770", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023771", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023772", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023773", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023774", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023775", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023776", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023777", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023778", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023779", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023780", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023781", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023782", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023783", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023784", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023785", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023786", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023787", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023788", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023789", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023790", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023791", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023792", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023793", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023794", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023795", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023796", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023797", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023798", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023799", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023800", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023801", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023802", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023803", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023804", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023805", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023806", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023807", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023808", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023809", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023810", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023811", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023812", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023813", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023816", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023818", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023819", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023820", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023821", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023822", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023823", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023824", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023825", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023826", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023827", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023828", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023829", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023830", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023831", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023832", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023833", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023834", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023835", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023836", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023837", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023838", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023839", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023840", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023841", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023842", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023843", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023844", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023845", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023846", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023847", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023848", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023849", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023850", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023851", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023852", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023853", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023854", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023855", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023856", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023857", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023858", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023859", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023860", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023861", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023862", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023863", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023864", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023865", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023866", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023867", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023868", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023869", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023870", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023871", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023872", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023873", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023874", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023875", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023876", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023877", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023878", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023879", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023880", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023881", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023882", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023883", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023884", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023885", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023886", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023887", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023888", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023889", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023890", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023891", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023892", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023893", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023894", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023895", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023896", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023897", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023898", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023899", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023900", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023901", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023902", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023903", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023904", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023905", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023906", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023907", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023908", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023909", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023910", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023911", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023912", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023913", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023914", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023915", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023916", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023917", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023918", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023919", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023920", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023921", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023922", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023923", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023924", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023925", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023926", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023927", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023928", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023929", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023930", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023931", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023932", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023933", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023934", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023935", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023936", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023937", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023938", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023939", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023940", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023941", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023942", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023943", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023944", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023945", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023946", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023947", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023948", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023949", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023950", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023951", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023952", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023953", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023954", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023955", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023956", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023957", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023958", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023959", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023960", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023961", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023962", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023963", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023964", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023965", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023966", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023967", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023968", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023969", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023970", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023971", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023972", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023973", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023974", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023975", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023976", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023977", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023978", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023979", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023980", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023981", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023982", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023983", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023984", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023985", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023986", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023987", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023988", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023989", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023990", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023991", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_023992", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023994", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_023995", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023996", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023998", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_023999", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024000", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024001", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024002", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024003", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024004", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024005", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024006", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024007", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024008", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024009", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024010", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024011", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024012", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024013", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024014", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024015", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024016", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024017", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024018", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024019", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024020", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024021", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024022", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024023", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024024", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024025", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024026", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024027", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024028", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024029", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024030", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024031", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024032", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024034", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024035", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024036", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024037", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024038", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024039", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024040", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024041", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024042", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024043", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024044", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024045", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024046", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024047", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024048", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024049", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024050", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024051", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024052", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024053", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024054", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024055", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024056", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024057", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024058", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024059", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024060", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024061", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024062", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024063", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024064", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024065", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024066", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024067", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024068", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024069", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024070", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024071", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024072", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024073", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024074", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024075", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024076", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024077", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024078", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024079", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024080", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024081", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024082", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024083", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024084", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024085", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024086", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024087", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024088", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024089", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024090", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024091", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024092", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024093", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024094", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024095", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024096", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024097", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024098", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024099", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024100", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024101", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024102", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024103", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024104", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024105", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024106", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024107", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024108", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024109", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024110", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024111", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024112", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024113", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024114", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024115", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024116", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024117", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024118", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024119", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024120", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024121", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024122", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024123", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024124", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024125", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024126", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024127", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024128", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024129", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024130", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024131", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024132", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024133", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024134", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024135", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024136", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024137", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024138", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024139", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024140", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024141", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024142", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024143", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024144", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024145", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024146", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024147", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024148", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024149", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024150", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024151", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024152", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024153", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024154", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024155", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024156", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024157", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024158", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024159", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024160", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024161", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024162", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024163", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024164", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024165", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024166", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024167", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024168", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024169", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024170", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024171", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024172", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024173", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024174", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024175", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024176", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024177", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024178", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024179", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024180", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024181", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024182", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024183", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024184", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024185", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024186", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024187", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024188", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024189", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024190", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024191", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024192", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024193", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024194", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024195", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024196", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024197", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024198", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024199", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024200", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024201", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024202", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024203", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024204", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024205", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024206", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024207", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024209", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024210", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024212", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024213", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024214", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024215", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024216", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024217", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024218", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024219", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024220", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024221", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024222", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024223", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024224", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024225", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024226", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024227", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024228", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024229", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024230", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024231", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024232", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024233", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024234", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024235", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024236", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024237", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024238", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024239", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024240", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024241", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024242", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024243", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024244", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024245", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024246", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024247", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024248", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024249", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024250", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024251", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024252", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024253", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024254", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024255", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024256", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024257", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024258", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024259", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024260", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024261", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024262", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024263", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024264", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024265", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024266", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024267", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024268", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024269", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024270", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024271", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024272", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024273", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024274", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024275", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024276", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024277", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024278", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024279", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024280", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024281", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024282", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024283", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024284", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024285", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024286", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024287", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024288", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024289", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024290", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024291", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024292", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024293", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024294", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024295", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024296", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024297", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024298", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024299", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024300", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024301", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024302", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024303", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024304", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024305", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024306", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024307", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024308", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024309", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024310", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024311", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024312", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024313", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024314", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024315", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024316", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024317", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024318", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024319", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024320", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024321", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024322", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024323", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024324", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024325", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024326", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024327", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024328", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024329", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024330", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024331", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024332", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024333", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024334", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024335", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024336", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024337", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024338", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024339", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024340", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024341", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024342", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024343", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024344", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024345", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024346", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024347", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024348", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024349", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024350", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024351", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024352", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024353", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024354", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024355", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024356", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024357", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024358", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024359", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024360", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024361", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024362", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024363", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024364", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024365", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024366", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024367", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024368", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024369", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024370", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024371", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024372", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024373", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024374", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024375", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024376", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024377", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024378", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024379", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024380", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024381", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024382", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024383", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024384", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024385", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024386", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024387", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024388", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024389", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024390", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024391", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024392", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024393", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024394", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024395", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024396", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024397", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024398", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024399", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024400", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024401", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024402", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024403", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024404", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024405", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024406", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024407", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024408", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024409", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024410", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024411", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024412", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024413", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024414", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024415", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024416", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024417", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024418", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024419", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024420", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024421", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024422", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024423", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024424", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024425", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024426", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024427", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024428", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024429", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024430", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024431", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024432", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024433", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024434", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024435", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024436", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024437", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024438", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024439", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024440", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024441", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024442", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024443", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024444", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024445", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024446", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024447", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024448", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024449", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024450", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024451", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024452", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024453", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024454", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024455", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024456", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024457", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024458", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024459", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024460", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024461", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024462", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024463", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024464", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024465", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024466", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024467", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024468", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024469", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024470", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024471", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024472", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024473", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024474", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024475", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024476", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024477", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024478", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024479", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024480", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024481", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024482", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024483", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024484", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024485", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024486", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024487", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024488", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024489", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024490", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024491", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024492", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024493", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024494", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024495", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024496", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024497", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024498", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024499", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024500", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024501", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024502", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024503", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024504", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024505", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024506", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024507", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024508", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024509", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024510", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024511", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024512", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024513", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024514", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024515", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024516", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024517", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024518", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024519", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024520", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024521", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024522", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024523", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024524", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024525", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024526", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024527", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024528", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024529", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024530", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024531", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024532", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024533", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024534", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024535", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024536", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024537", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024538", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024539", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024540", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024541", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024542", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024543", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024544", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024545", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024546", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024547", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024548", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024549", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024550", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024551", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024552", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024553", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024554", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024555", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024556", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024557", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024558", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024559", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024560", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024561", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024562", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024563", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024564", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024565", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024566", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024567", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024568", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024569", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024570", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024571", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024572", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024573", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024574", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024575", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024576", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024577", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024578", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024579", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024580", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024581", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024582", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024583", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024584", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024585", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024586", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024587", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024588", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024589", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024590", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024591", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024592", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024593", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024594", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024595", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024596", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024597", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024598", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024599", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024600", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024601", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024602", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024603", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024604", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024605", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024606", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024607", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024608", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024609", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024610", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024611", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024612", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024613", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024614", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024615", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024616", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024617", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024618", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024619", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024620", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024621", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024622", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024623", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024624", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024625", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024626", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024627", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024628", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024629", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024630", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024631", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024632", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024633", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024634", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024635", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024636", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024637", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024638", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024639", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024640", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024641", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024642", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024643", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024644", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024645", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024646", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024647", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024648", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024649", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024650", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024651", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024652", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024653", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024654", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024655", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024656", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024657", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024658", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024659", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024660", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024661", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024662", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024663", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024664", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024665", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024666", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024667", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024668", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024669", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024670", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024671", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024672", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024673", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024674", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024675", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024676", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024677", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024678", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024679", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024680", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024681", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024682", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024683", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024684", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024685", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024686", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024687", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024688", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024689", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024690", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024691", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024692", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024693", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024694", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024695", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024696", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024697", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024698", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024699", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024700", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024701", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024702", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024703", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024704", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024705", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024706", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024707", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024708", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024709", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024710", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024713", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024715", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024718", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024719", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024720", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024721", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024722", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024723", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024724", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024725", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024726", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024727", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024728", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024729", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024730", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024731", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024732", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024733", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024735", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024736", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024737", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024738", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024739", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024740", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024741", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024742", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024743", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024744", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024745", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024746", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024747", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024748", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024749", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024750", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024751", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024752", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024753", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024754", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024755", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024756", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024757", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024758", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024759", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024760", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024761", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024762", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024763", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024764", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024765", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024766", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024767", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024768", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024769", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024770", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024771", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024772", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024773", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024774", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024775", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024776", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024777", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024778", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024779", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024780", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024781", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024782", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024783", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024784", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024785", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024786", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024787", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024788", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024789", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024790", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024791", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024792", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024793", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024794", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024795", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024796", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024797", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024798", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024799", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024800", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024801", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024802", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024803", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024804", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024805", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024806", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024807", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024808", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024809", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024810", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024811", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024812", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024813", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024814", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024815", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024816", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024817", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024818", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024819", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024820", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024821", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024822", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024823", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024824", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024825", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024826", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024827", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024828", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024829", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024830", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024831", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024832", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024833", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024834", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024835", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024836", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024837", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024838", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024839", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024840", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024841", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024842", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024843", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024844", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024845", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024846", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024847", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024848", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024849", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024850", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024851", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024852", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024853", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024854", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024855", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024856", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024857", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024858", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024859", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024860", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024861", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024862", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024863", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024864", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024865", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024866", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024867", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024868", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024869", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024870", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024871", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024872", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024873", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024874", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024875", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024876", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024877", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024878", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024879", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024880", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024881", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024882", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024883", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024884", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024885", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024886", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024887", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024888", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024889", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024890", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024891", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024892", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024893", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024894", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024895", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024896", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
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{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024898", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024899", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024900", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024901", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024902", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024903", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024904", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024905", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024906", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024907", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024908", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024909", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024910", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024911", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024912", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024913", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024914", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024915", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024916", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024917", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024918", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024919", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024920", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024921", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024922", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024923", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024924", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024925", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024926", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024927", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024928", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024929", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024930", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024931", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024932", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024933", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024934", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024935", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024936", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024937", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024938", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024939", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024940", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024941", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024942", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024943", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024944", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024945", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024946", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024947", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024948", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024949", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024950", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024951", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024952", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024953", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024954", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024955", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024956", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024957", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024958", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024959", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024960", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024961", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024962", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Randomized controlled trial? Provide a precise definition and one implication."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024963", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024964", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024965", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024966", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024967", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024968", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024969", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type I error in 1–2 sentences for an advanced academic audience."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024970", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024971", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024972", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Posterior concisely and include one example application."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024973", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024974", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024975", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Randomized controlled trial in 1–2 sentences for an advanced academic audience."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024976", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024977", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type I error concisely and include one example application."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024978", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024979", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024980", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Homoskedasticity concisely and include one example application."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024981", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Type II error in 1–2 sentences for an advanced academic audience."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024982", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Multiple comparisons problem? Provide a precise definition and one implication."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024983", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024984", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type II error? Provide a precise definition and one implication."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024985", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024986", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024987", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024988", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Posterior? Provide a precise definition and one implication."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024989", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Type II error concisely and include one example application."}]}, "response": "Failing to reject a false null; related to power (1-β). Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024990", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024991", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Multiple comparisons problem concisely and include one example application."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024992", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024993", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Type I error? Provide a precise definition and one implication."}]}, "response": "Rejecting a true null; probability controlled by α. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024994", "domain": "master_scholars_academics", "subdomain": "stats::inference", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "What is Homoskedasticity? Provide a precise definition and one implication."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024995", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Posterior in 1–2 sentences for an advanced academic audience."}]}, "response": "Updated distribution after observing data; proportional to prior × likelihood. Implication: it constrains correct interpretation under assumptions.", "meta": {"id": "WUA_STAT_25K_024996", "domain": "master_scholars_academics", "subdomain": "stats::bayes", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024997", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Multiple comparisons problem in 1–2 sentences for an advanced academic audience."}]}, "response": "Testing many hypotheses inflates false positives unless controlled (e.g., FDR). Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_024998", "domain": "master_scholars_academics", "subdomain": "stats::robustness", "task": "definition", "difficulty": "seed", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Define Homoskedasticity in 1–2 sentences for an advanced academic audience."}]}, "response": "Assumption of constant error variance across predictor values in regression. Example: used in applied analysis and research reporting.", "meta": {"id": "WUA_STAT_25K_024999", "domain": "master_scholars_academics", "subdomain": "stats::regression", "task": "definition", "difficulty": "advanced", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}
{"prompt": {"messages": [{"role": "system", "content": "You are WithinUsAI Master Scholars Statistics. Provide precise definitions; avoid fluff."}, {"role": "user", "content": "Explain Randomized controlled trial concisely and include one example application."}]}, "response": "Experiment with randomized assignment to support causal inference. Implication: it supports unit-aware quantitative modeling.", "meta": {"id": "WUA_STAT_25K_025000", "domain": "master_scholars_academics", "subdomain": "stats::design", "task": "definition", "difficulty": "core", "truth_mode": "textbook_definition", "wrapper": "micro_refresher", "created_utc": "2026-01-04T05:02:58Z", "license": "cc-by-4.0", "tags": ["statistics", "academics", "definitions"]}, "fact_sources": [{"source_title": "OpenStax Introductory Statistics (CC BY 4.0)", "source_url": "https://openstax.org/details/books/introductory-statistics", "evidence_type": "standard", "accessed_at_utc": "2026-01-04T05:02:58Z"}, {"source_title": "NIST/SEMATECH e-Handbook of Statistical Methods", "source_url": "https://www.itl.nist.gov/div898/handbook/", "evidence_type": "official", "accessed_at_utc": "2026-01-04T05:02:58Z"}], "artifacts": {"max_tokens_hint": 220, "style_contract": "concise_formal"}}